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5.1: Rules of Exponents

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• Page ID 18354

Learning Objectives

• Simplify expressions using the rules of exponents.
• Simplify expressions involving parentheses and exponents.
• Simplify expressions involving \(0\) as an exponent.

Product, Quotient, and Power Rule for Exponents

If a factor is repeated multiple times, then the product can be written in exponential form \(x_{n}\). The positive integer exponent \(n\) indicates the number of times the base \(x\) is repeated as a factor

For example,

\(5^{4}=5\cdot 5\cdot 5\cdot 5\)

Here the base is \(5\) and the exponent is \(4\). Exponents are sometimes indicated with the caret (^) symbol found on the keyboard: \(5\)^\(4 = 5*5*5*5\).

Next consider the product of \(2^{3}\) and \(2^{5}\),

Expanding the expression using the definition produces multiple factors of the base, which is quite cumbersome, particularly when \(n\) is large. For this reason, we will develop some useful rules to help us simplify expressions with exponents. In this example, notice that we could obtain the same result by adding the exponents.

\(2^{3}\cdot 2^{5}=2^{3+5}=2^{8}\)

In general, this describes the product rule for exponents. If \(m\) and \(n\) are positive integers, then

\[x^{m}\cdot x^{n} = x^{m+n}\]

In other words, when multiplying two expressions with the same base, add the exponents.

Example \(\PageIndex{1}\)

Simplify: \(10^{5}\cdot 10^{18}\).

\(\begin{aligned} 10^{5}\cdot 10^{18}&=10^{5+18} \\ &=10^{23} \end{aligned}\)

\(10^{23}\)

In the previous example, notice that we did not multiply the base 10 times itself. When applying the product rule, add the exponents and leave the base unchanged.

Example \(\PageIndex{2}\)

Simplify: \(x^{6}⋅x^{12}⋅x\).

Recall that the variable \(x\) is assumed to have an exponent of \(1: x=x^{1}\).

\(\begin{aligned} x^{6}\cdot x^{12}\cdot x &=x^{6}\cdot x^{12}\cdot x^{1} \\ &=x^{6+12+1} \\ &=x^{19} \end{aligned}\)

The base could be any algebraic expression.

Example \(\PageIndex{3}\)

Simplify: \((x+y)^{9} (x+y)^{13}\).

Treat the expression \((x+y)\) as the base.

\(\begin{aligned} (x+y)^{9}(x+y)^{13}&=(x+y)^{9+13} \\ &=(x+y)^{22} \end{aligned}\)

\((x+y)^{22}\)

The commutative property of multiplication allows us to use the product rule for exponents to simplify factors of an algebraic expression.

Example \(\PageIndex{4}\)

Simplify: \(2x^{8}y⋅3x^{4}y^{7}\).

Multiply the coefficients and add the exponents of variable factors with the same base.

\(\begin{aligned} 2x^{8}y\cdot 3x^{4}y^{7}&=2\cdot 3\cdot x^{8}\cdot x^{4}\cdot y^{1}\cdot y^{7} &\color{Cerulean}{Commutative\:property} \\ &=6\cdot x^{8+4}\cdot y^{1+7} &\color{Cerulean}{Power\:rule\:for\:exponents} \\ &=6x^{12}y^{8} \end{aligned}\)

\(6x^{12}y^{8}\)

Next, we will develop a rule for division by first looking at the quotient of \(2^{7}\) and \(2^{3}\).

Here we can cancel factors after applying the definition of exponents. Notice that the same result can be obtained by subtracting the exponents.

\[\frac{2^{7}}{2^{3}}=2^{7-3}=2^{4} \nonumber\]

This describes the quotient rule for exponents. If \(m\) and \(n\) are positive integers and \(x≠0\), then

\[\frac{x^{m}}{x^{n}}=x^{m-n} \nonumber\]

In other words, when you divide two expressions with the same base, subtract the exponents.

Example \(\PageIndex{5}\)

Simplify: \(\frac{12y^{15}}{4y^{7}}\).

Divide the coefficients and subtract the exponents of the variable \(y\).

\(\begin{aligned} \frac{12y^{15}}{4y^{7}}&=\frac{12}{4}\cdot y^{15-7}\\ &=3y^{8} \end{aligned}\)

Example \(\PageIndex{6}\)

Simplify: \(\frac{20x^{10}(x+5)^{6}}{10x^{9}(x+5)^{2}}\)

\(\begin{aligned} \frac{20x^{10}(x+5)^{6}}{10x^{9}(x+5)^{2}}&=\frac{20}{10}\cdot x^{10-9}\cdot (x+5)^{6-2} \\ &=2x^{1}(x+5)^{4} \end{aligned}\)

\(2x(x+5)^{4}\)

Now raise \(2^{3}\) to the fourth power as follows:

After writing the base \(2^{3}\) as a factor four times, expand to obtain \(12\) factors of \(2\). We can obtain the same result by multiplying the exponents.

\((2^{3})^{^{4}} = 2^{3\cdot 4} = 2^{12}\)

In general, this describes the power rule for exponents. Given positive integers \(m\) and \(n\), then

\[(x^{m})^{^{n}}=x^{m\cdot n}\]

In other words, when raising a power to a power, multiply the exponents.

Example \(\PageIndex{7}\)

Simplify: \((y^{6})^{^{7}}=y^{6\cdot 7}\)

\(\begin{aligned} (y^{6})^{^{7}}&=y^{6\cdot 7} \\ &=y^{42} \end{aligned}\)

To summarize, we have developed three very useful rules of exponents that are used extensively in algebra. If given positive integers \(m\) and \(n\), then

• Product rule: \[x^{m}\cdot x^{n}=x^{m+n}\]
• Quotient rule: \[\frac{x^{m}}{x^{n}}=x^{m-n}, x\neq 0\]
• Power rule: \[(x^{m})^{^{n}} = x^{m\cdot n}\]

Exercise \(\PageIndex{1}\)

Simplify: \(y^{5}⋅(y^{4})^{^{6}}\).

Power Rules for Products and Quotients

Now we consider raising grouped products to a power. For example,

\(\begin{aligned} (xy)^{4} &= xy\cdot xy\cdot xy\cdot xy \\ &=x\cdot x\cdot x\cdot x\cdot y\cdot y\cdot y\cdot y\quad\color{Cerulean}{Commutative\:property} \\ &=x^{4}\cdot y^{4} \end{aligned}\)

After expanding, we have four factors of the product \(xy\). This is equivalent to raising each of the original factors to the fourth power. In general, this describes the power rule for a product. If \(n\) is a positive integer, then

\[(xy)^{n}=x^{n}y^{n}\]

Example \(\PageIndex{8}\)

Simplify: \((2ab)^{7}=2^{7}a^{7}b^{7}\).

We must apply the exponent \(7\) to all the factors, including the coefficient, \(2\).

\(\begin{aligned} (2ab)^{7}&=2^{7}a^{7}b^{7} \\ &=128a^{7}b^{7} \end{aligned}\)

If a coefficient is raised to a relatively small power, then present the real number equivalent, as we did in this example: \(2^{7}=128\).

\(128a^{7}b^{7}\)

In many cases, the process of simplifying expressions involving exponents requires the use of several rules of exponents.

Example \(\PageIndex{9}\)

Simplify: \((3xy^{3})^{^{4}}\).

\(\begin{aligned} (3xy^{3})^{^{4}}&=3^{4}\cdot x^{4}\cdot (y^{3})^{^{4}} &\color{Cerulean}{Power\:rule\:for\:products} \\ &=3^{4}x^{4}y^{3\cdot 4} &\color{Cerulean}{Power\:rule\:for\:exponents} \\ &=81x^{4}y^{12} \end{aligned}\)

\(81x^{4}y^{12}\)

Example \(\PageIndex{10}\)

Simplify: \((4x^{2}y^{5}z)^{^{3}}\).

\(\begin{aligned} (4x^{2}y^{5}z)^{^{3}}&=4^{3}\cdot(x^{2})^{^{3}}\cdot (y^{5})^{^{3}}\cdot z^{3} \\ &=64x^{6}y^{15}z^{3} \end{aligned}\)

\(64x^{6}y^{15}z^{3}\)

Example \(\PageIndex{11}\)

Simplify: \([5(x+y)^{3}]^{^{3}}\)

\(\begin{aligned} [5(x+y)^{3}]^{^{3}} &=5^{3}\cdot (x+y)^{9} \\ &=125(x+y)^{9} \end{aligned}\)

\(125(x+y)^{9}\)

Next, consider a quotient raised to a power.

\(\begin{aligned} \left( \frac{x}{y} \right) ^{4} &= \frac{x}{y}\cdot \frac{x}{y}\cdot \frac{x}{y}\cdot \frac{x}{y} \\ &=\frac{x\cdot x\cdot x\cdot x}{y\cdot y\cdot y\cdot y} \\ &=\frac{x^{4}}{y^{4}} \end{aligned}\)

Here we obtain four factors of the quotient, which is equivalent to the numerator and the denominator both raised to the fourth power. In general, this describes the power rule for a quotient. If \(n\) is a positive integer and \(y≠0\), then

\[\left( \frac{x}{y} \right) ^{n} = \frac{x^{n}}{y^{n}}\]

In other words, given a fraction raised to a power, we can apply that exponent to the numerator and the denominator. This rule requires that the denominator is nonzero. We will make this assumption for the remainder of the section.

Example \(\PageIndex{12}\)

Simplify: \(\left(\frac{3a}{b} \right) ^{3}\)

First, apply the power rule for a quotient and then the power rule for a product.

\(\begin{aligned} \left(\frac{3a}{b} \right) ^{3}&=\frac{(3a)^{3}}{b^{3}} &\color{Cerulean}{Power\:rule\:for\:a\:quotient} \\ &=\frac{3^{3}\cdot a^{3}}{b^{3}} &\color{Cerulean}{Power\:rule\:for\:a\:product} \\ &=\frac{27a^{3}}{b^{3}} \end{aligned}\)

\(\frac{27a^{3}}{b^{3}}\)

In practice, we often combine these two steps by applying the exponent to all factors in the numerator and the denominator.

Example \(\PageIndex{13}\)

Simplify: \(\left( \frac{ab^{2}}{2c^{3}} \right)^{5}\)

Apply the exponent \(5\) to all of the factors in the numerator and the denominator.

\(\begin{aligned} \left( \frac{ab^{2}}{2c^{3}} \right)^{5}&=\frac{a^{5}(b^{2})^{^{5}}}{2^{5}(c^{3})^{^{5}}} \\ &=\frac{a^{5}b^{10}}{32c^{15}} \end{aligned}\)

\(\frac{a^{5}b^{10}}{32c^{15}}\)

Example \(\PageIndex{14}\)

Simplify: \(\left( \frac{5x^{5}(2x-1)^{4}}{3y^{7}} \right) ^{2}\)

\(\begin{aligned} \left( \frac{5x^{5}(2x-1)^{4}}{3y^{7}} \right) ^{2} &=\frac{(5x^{5}(2x-1)^{4})^{2}}{(3y^{7})^{^{2}}} &\color{Cerulean}{Power\:rule\:for\:a\:quotient} \\ &=\frac{5^{2}\cdot (x^{5})^{^{2}}\cdot [(2x-1)^{4}]^{2}}{3^{2}\cdot (y^{7})^{^{2}}} &\color{Cerulean}{Power\:rule\:for\:products} \\ &=\frac{25x^{10}(2x-1)^{8}}{9y^{14}} &\color{Cerulean}{Power\:rule\:for\:exponents} \end{aligned}\)

\(\frac{25x^{10}(2x-1)^{8}}{9y^{14}}\)

It is a good practice to simplify within parentheses before using the power rules; this is consistent with the order of operations.

Example \(\PageIndex{15}\)

Simplify: \(\left( \frac{-2x^{3}y^{4}z}{xy^{2}} \right)^{4}\)

\(\begin{aligned} \left( \frac{-2x^{3}y^{4}z}{xy^{2}} \right)^{4}&=(-2\cdot x^{3-1}\cdot y^{4-2}\cdot z)^{4} &\color{Cerulean}{Simplify\:within\:the\:parentheses\:first.} \\ &=(-2\cdot x^{2}\cdot y^{2} \cdot z)^{4} &\color{Cerulean}{Apply\:the\:power\:rule\:for\:a\:product.} \\ &=(-2)^{4}\cdot (x^{2})^{^{4}}\cdot (y^{2})^{^{4}}\cdot z^{4}&\color{Cerulean}{Apply\:the\:power\:rule\:for\:exponents.} \\ &=16x^{8}y^{8}z^{4} \end{aligned}\)

\(16x^{8}y^{8}z^{4}\)

To summarize, we have developed two new rules that are useful when grouping symbols are used in conjunction with exponents. If given a positive integer \(n\), where \(y\) is a nonzero number, then

• Power rule for a product: \[(xy)^{n} = x^{n}y^{n}\]
• Power rule for a quotient: \[\left( \frac{x}{y} \right)^{n} = \frac{x^{n}}{y^{n}}\]

Exercise \(\PageIndex{2}\)

Simplify: \(\left(\frac{4x^{2}(x-y)^{3}}{3yz^{5}} \right)^{3}\)

\(\frac{64x^{6}(x-y)^{9}}{27y^{3}z^{15}}\)

Zero as an Exponent

Using the quotient rule for exponents, we can define what it means to have \(0\) as an exponent. Consider the following calculation:

\(\color{Cerulean}{1}\color{black}{=\frac{8}{8}=\frac{2^{3}}{2^{3}}=2^{3-3}=}\color{Cerulean}{2^{0}}

Eight divided by \(8\) is clearly equal to \(1\), and when the quotient rule for exponents is applied, we see that a \(0\) exponent results. This leads us to the definition of zero as an exponent, where \(x≠0\):

\[x^{0}=1\]

It is important to note that \(0^{0}\) is undefined. If the base is negative, then the result is still \(+1\). In other words, any nonzero base raised to the \(0\) power is defined to be \(1\). In the following examples, assume all variables are nonzero.

Example \(\PageIndex{16}\)

• \((-5)^{0}\)
• Any nonzero quantity raised to the \(0\) power is equal to \(1\).

\((-5)^{0}=1\)

b.In the example \(−5^{0}\), the base is \(5\), not \(−5\).

Example \(\PageIndex{17}\)

\((5x^{3}y^{0}z^{2})^{^{2}}\).

It is good practice to simplify within the parentheses first.

\(\begin{aligned} (5x^{3}\color{Cerulean}{y^{0}}\color{black}{z^{2})^{^{2}}}&=(5x^{3}\cdot\color{Cerulean}{1}\color{black}{\cdot z^{2})^{2}} \\ &=(5x^{3}z^{2})^{2} \\ &=5^{2}x^{3\cdot 2}z^{2\cdot 2} \\ &=25x^{6}z^{4} \end{aligned}\)

\(25x^{6}z^{4}\)

Example \(\PageIndex{18}\)

\(\left( -\frac{8a^{10}b^{5}}{5c^{12}d^{14}} \right) ^{0}\).

\(\left( -\frac{8a^{10}b^{5}}{5c^{12}d^{14}} \right) ^{0} =1\)

Exercise \(\PageIndex{3}\)

\(5x^{0}\) and \((5x)^{0}\)

\(5x^{0}=5\) and \((5x)^{0}=1\)

Key Takeaways

• The rules of exponents allow you to simplify expressions involving exponents.
• When multiplying two quantities with the same base, add exponents: \(x^{m}⋅x^{n}=x^{m+n}\).
• When dividing two quantities with the same base, subtract exponents: \(\frac{x^{m}}{x^{n}}=x^{m−n}\).
• When raising powers to powers, multiply exponents: \((x^{m})^{^{n}}=x^{m⋅n}\).
• When a grouped quantity involving multiplication and division is raised to a power, apply that power to all of the factors in the numerator and the denominator: \((xy)^{n}=x^{n}y^{n}\) and \((\frac{x}{y})^{n}=\frac{x^{n}}{y^{n}}\).
• Any nonzero quantity raised to the \(0\) power is defined to be equal to \(1: x^{0}=1\).

Exercise \(\PageIndex{4}\) Product, Quotient, and Power Rule for Exponents

Write each expression using exponential form.

• \((2x)(2x)(2x)(2x)(2x)\)
• \((−3y)(−3y)(−3y)\)
• \(−10⋅a⋅a⋅a⋅a⋅a⋅a⋅a\)
• \(12⋅x⋅x⋅y⋅y⋅y⋅y⋅y⋅y\)
• \(−6⋅(x−1)(x−1)(x−1)\)
• \((9ab)(9ab)(9ab)(a^{2}−b)(a^{2}−b)\)

1. \((2x)^{5}\)

3. \(-10a^{7}\)

5. \(-6(x-1)^{3}\)

Exercise \(\PageIndex{5}\) Product, Quotient, and Power Rule for Exponents

• \(2^{7}⋅2^{5}\)
• \(3^{9}⋅3\)
• \(−2^{4}\)
• \((−2)^{4}\)
• \(−3^{3}\)
• \((−3)^{4}\)
• \(10^{13}⋅10^{5}⋅10^{4}\)
• \(10^{8}⋅10^{7}⋅10\)
• \(\frac{5^{12}}{5^{2}}\)
• \(\frac{10^{7}}{10^{10}}\)
• \(\frac{10^{12}}{10^{9}}\)
• \((7^{3})^{^{5}}\)
• \((4^{8})^{^{4}}\)
• \(10^{6}⋅(10^{5})^{^{4}}\)

1. \(2^{12}\)

3. \(−16\)

5. \(−27\)

7. \(10^{22}\)

9. \(5^{10}\)

11. \(10^{3}\)

13. \(4^{32}\)

Exercise \(\PageIndex{6}\) Product, Quotient, and Power Rule for Exponents

• \((−x)^{6}\)
• \(a^{5}⋅(−a)^{2}\)
• \(x^{3}⋅x^{5}⋅x\)
• \(y^{5}⋅y^{4}⋅y^{2}\)
• \((a^{5})^{^{2}}⋅(a^{3})^{^{4}}⋅a\)
• \((x+1)^{4}(y^{5})^{^{4}}⋅y^{2}\)
• \((x+1)^{5}(x+1)^{8}\)
• \((2a−b)^{12}(2a−b)^{9}\)
• \(\frac{(3x-1)^{5}}{(3x-1)^{2}}\)
• \(\frac{(a-5)^{37}}{(a-5)^{13}}\)
• \(xy^{2}⋅x^{2}y\)
• \(3x^{2}y^{3}⋅7xy^{5}\)
• \(−8a^{2}b⋅2ab\)
• \(−3ab^{2}c^{3}⋅9a^{4}b^{5}c^{6}\)
• \(2a^{2}b^{4}c (−3abc)\)
• \(5a^{2}(b^{3})^{^{3}}c^{3}⋅(−2)2a^{3}(b^{2})^{^{4}}\)
• \(2x^{2}(x+y)^{5}⋅3x^{5}(x+y)^{4}\)
• \(−5xy^{6}(2x−1)^{6}⋅x^{5}y(2x−1)^{3}\)
• \(x^{2}y⋅xy^{3}⋅x^{5}y^{5}\)
• \(−2x^{10}y⋅3x^{2}y^{12}⋅5xy^{3}\)
• \(3^{2}x^{4}y^{2}z⋅3xy^{4}z^{4}\)
• \((−x^{2})^{^{3}}(x^{3})^{^{2}}(x^{4})^{^{3}}\)
• \(a^{10}⋅\frac{(a^{6})^{^{3}}}{a^{3}}\)
• \(\frac{10x^{9}(x^{3})^{^{5}}}{2x^{5}}\)
• \(\frac{a^{6}b^{3}}{a^{2}b^{2}}\)
• \(\frac{m^{10}n^{7}}{m^{3}n^{4}}\)
• \(\frac{20x^{5}y^{12}z^{3}}{10x^{2}y^{10}z}\)
• \(\frac{-24a^{16}b^{12}c^{3}}{6a^{6}b^{11}c}\)
• \(\frac{16x^{4}(x+2)^{3}}{4x(x+2)}\)
• \(\frac{50y^{2}(x+y)^{20}}{10y(x+y)^{17}}\)

1. \(x^{6}\)

3. \(x^{9}\)

5. \(a^{23}\)

7. \((x+1)^{13}\)

9. \((3x−1)^{3}\)

11. \(x^{3}y^{3}\)

13. \(−16a^{3}b^{2}\)

15. \(−6a^{3}b^{5}c^{2}\)

17. \(6x^{7}(x+y)^{9}\)

19. \(x^{8}y^{9}\)

21. \(27x^{5}y^{6}z^{5}\)

23. \(a^{25}\)

25. \(a^{4}b\)

27. \(2x^{3}y^{2}z^{2}\)

29. \(4x^{3}(x+2)^{2}\)

Exercise \(\PageIndex{7}\) Power Rules for Products and Quotients

• \((2x)^{5}\)
• \((−3y)^{4}\)
• \((−xy)^{3}\)
• \((5xy)^{3}\)
• \((−4abc)^{2}\)
• \(\left(\frac{7}{2x} \right)^{2}\)
• \(-\left(\frac{5}{3y} \right)^{3}\)
• \((3abc)^{3}\)
• \(\left(\frac{-2xy}{3z} \right)^{4}\)
• \(\left(\frac{5y}{(2x-1)x}\right)^{3}\)
• \((3x^{2})^{^{3}}\)
• \((−2x^{3})^{^{2}}\)
• \((xy^{5})^{^{7}}\)
• \((x^{2}y^{10})^{^{2}}\)
• \(\left(\frac{3x^{2}}{y} \right)^{3}\)
• \((2x^{2}y^{3}z^{4})^{^{5}}\)
• \(\left(\frac{-7ab^{4}}{c^{2}} \right)^{2}\)
• \([x^{5}y^{4}(x+y)^{4}]^{5}\)
• \([2y(x+1)^{5}]^{3}\)
• \((ab^{3})^{^{3}}\)
• \(\left(\frac{5a^{2}}{3b} \right)^{4}\)
• \(\left(\frac{-2x^{3}}{3y^{2}} \right)^{2}\)
• \(\left(\frac{-x^{2}}{y^{3}} \right)^{3}\)
• \(\left(\frac{ab^{2}}{3c^{3}d^{2}} \right)^{4}\)
• \(\left(\frac{2x^{7}y}{(x-1)^{3}z^{5}} \right)^{6}\)
• \((2x^{4})^{^{3}}⋅(x^{5})^{^{2}}\)
• \((x^{3}y)^{^{2}}⋅(xy^{4})^{^{3}}\)
• \((−2a^{2}b^{3})^{^{2}}⋅(2a^{5}b)^{^{4}}\)
• \((−a^{2}b)^{3}(3ab^{4})^{4}\)
• \((2x^{3}(x+y)^{4})^{5}⋅(2x^{4}(x+y)^{2})^{3}\)
• \(\left(\frac{-3x^{5}y^{4}}{xy^{2}} \right)^{3}\)
• \(\left(\frac{-3x^{5}y^{4}}{xy^{2}} \right)^{2}\)
• \(\left(\frac{-25x^{10}y^{15}}{5x^{5}y^{10}} \right)^{3}\)
• \(\left(\frac{10x^{3}y^{5}}{5xy^{2}} \right)^{2}\)
• \(\left(\frac{-24ab^{3}}{6bc} \right)^{5}\)
• \(\left(\frac{-2x^{3}y^{16}}{x^{2}y} \right)^{2}\)
• \(\left(\frac{30ab^{3}}{3abc} \right)^{3}\)
• \(\left(\frac{3s^{3}t^{2}}{2s^{2}t} \right)^{3}\)
• \(\left(\frac{6xy^{5}(x+y)^{6}}{3y^{2}z(x+y)^{2}} \right)^{5}\)
• \(\left(\frac{-64a^{5}b^{12}c^{2}(2ab-1)^{14}}{32a^{2}b^{10}c^{2}(2ab-1)^{7}} \right)^{4}\)
• The probability of tossing a fair coin and obtaining \(n\) heads in a row is given by the formula \(P=(12)^{n}\). Determine the probability, as a percent, of tossing \(5\) heads in a row.
• The probability of rolling a single fair six-sided die and obtaining \(n\) of the same faces up in a row is given by the formula \(P=(16)^{n}\). Determine the probability, as a percent, of obtaining the same face up two times in a row.
• If each side of a square measures \(2x^{3}\) units, then determine the area in terms of the variable \(x\).
• If each edge of a cube measures \(5x^{2}\) units, then determine the volume in terms of the variable \(x\).

1. \(32x^{5}\)

3. \(−x^{3}y^{3}\)

5. \(16a^{2}b^{2}c^{2}\)

7. \(−\frac{125}{27y^{3}}\)

9. \(\frac{16x^{4}y^{4}}{81z^{4}}\)

11. \(27x^{6}\)

13. \(x^{7}y^{35}\)

15. \(\frac{27x^{6}}{y^{3}}\)

17. \(\frac{49a^{2}b^{8}}{c^{4}}\)

19. \(8y^{3}(x+1)^{15}\)

21. \(\frac{625a^{8}}{81b^{4}}\)

23. \(−\frac{x^{6}}{y^{9}}\)

25. \(\frac{64x^{42}y^{6}}{(x−1)^{18}z^{30}}\)

27. \(x^{9}y^{14}\)

29. \(−81a^{10}b^{19}\)

31. \(−27x^{12}y^{6}\)

33. \(−125x^{15}y^{15}\)

35. \(\frac{−1024a^{5}b^{10}}{c^{5}}\)

37. \(\frac{1000b^{6}}{c^{3}}\)

39. \(\frac{32x^{5}y^{15}(x+y)^{20}}{z^{5}}\)

41. \(3 \frac{1}{8}\)%

43. \(A=4x^{6}\)

Exercise \(\PageIndex{8}\) Zero Exponents

Simplify. (Assume variables are nonzero.)

• \((−7)^{0}\)
• \(−10^{0}\)
• \(−3^{0}⋅(−7)^{0}\)
• \(8675309^{0}\)
• \(5^{2}⋅3^{0}⋅2^{3}\)
• \(−3^{0}⋅(−2)^{2}⋅(−3)^{0}\)
• \(\frac{5x^{0}}{y^{2}}\)
• \((−3)^{2}x^{2}y^{0}z^{5}\)
• \(−3^{2}(x^{3})^{2}y^{2}(z^{3})^{0}\)
• \(2x^{3}y^{0}z⋅3x^{0}y^{3}z^{5}\)
• \(−3ab^{2}c^{0}⋅3a^{2}(b^{3}c^{2})^{0}\)
• \((−8xy^{2})^{0}\)
• \((2x^{2}y^{3})^{0}\)
• \(\frac{9x^{0}y^{4}}{3y^{3}}\)

3. \(−1\)

7. \(−4\)

9. \(9x^{2}z^{5}\)

11. \(6x^{3}y^{3}z^{6}\)

Exercise \(\PageIndex{9}\) Discussion Board Topics

• René Descartes (1637) established the usage of exponential form: \(a^{2}, a^{3}\), and so on. Before this, how were exponents denoted?
• Discuss the accomplishments accredited to Al-Karismi.
• Why is \(0^{0}\) undefined?
• Explain to a beginning student why \(3^{4}⋅3^{2}≠9^{6}\).

1. Answers may vary

3. Answers may vary

Simplifying Exponents

Simplifying exponents is a method of simplifying the algebraic expressions involving exponents into a simpler form such that they cannot further be simplified. There are rules in algebra for simplifying exponents with different and same bases that we can use. Various arithmetic operations like addition, subtraction, multiplication, and division can be applied to simplify exponent algebraic expressions, exponents in fractions, and negative exponents using the laws of exponents.

In this article, we will learn how to simplify exponents in algebraic expressions, fractions, negative exponents, and with different bases using the simplifying exponents' rules. We will also solve various examples related to the concept for a better understanding.

What is Simplifying Exponents?

Let us first recall the concept of exponents before learning to simplify exponents . The exponent of a number shows how many times the number is multiplied by itself. When we apply arithmetic operations on exponents, we use the laws of exponents for simplifying exponent expressions. Simplifying exponents is a simple process of reducing the mathematical expressions involving exponents into a simpler form such that they cannot further be simplified. Let us first go through some of the important rules for simplifying exponents in the next section.

Simplifying Exponents Rules

Given below is a list of rules that we for simplifying exponents in algebraic expressions . These rules are also known as the laws of exponents and are named as per the operation involved. Let us have a look at these rules that we will use later for simplifying exponents:

• Product Rule : a m × a n = a m+n
• Quotient Rule : a m /a n = a m-n
• Zero Exponent Rule: a 0 = 1
• Identity Exponent Rule: a 1 = a
• Negative Exponents Rule: a -m = 1/a m ; (a/b) -m = (b/a) m
• Power of a Power Rule : (a m ) n = a mn
• Power of a Product Rule: (ab) m = a m b m
• Power of a Quotient Rule: (a/b) m = a m /b m

Simplifying Exponents With Different Bases

When we multiply exponents or divide exponents with different bases, there can be two cases: i) when the exponents are the same, ii) when the exponents are different. Let us discuss each of these cases and understand the process of simplifying exponents in such cases with the help of examples.

Simplifying Exponents With Different Bases and Same Power

When simplifying exponents with different bases and the same power, we follow the rules:

• a m × b m = (ab) m
• a m ÷ b m = (a÷b) m

Let us simplify the following exponents: i) 2 4 × 3 4 , ii) 4 3 ÷ 2 3

i) 2 4 × 3 4

ii) 4 3 ÷ 2 3

Simplifying Exponents With Different Bases and Different Power

When we have to simplify exponents with different bases and different power, we simplify the terms separately and then apply the arithmetic operation involved. Let us solve an example to understand this better. Simplify 23 × 52. Now, here the bases and powers both are different. So, for simplifying exponents in this expression, we will simplify the terms separately first.

Simplifying Exponents In Fractions

In this section, we will learn how to simplify exponents in fractions. When we are given algebraic expressions in fractions, we use the laws of exponents to simplify them. Let us understand this with the help of a few examples solved below:

Example 1 : Simplify (35x 3 y 2 z) / (7xy 4 )

Solution: We will simplify the given algebraic expression, using the simplifying exponents rules discussed above. So, we have

(35x 3 y 2 z) / (7xy 4 )

= (35/7) (x 3 /x) (y 2 /y 4 ) (z)

= 5 × x 3-1 × y 2-4 × z

= 5x 2 y -2 z

Answer: (35x 3 y 2 z) / (7xy 4 ) = 5x 2 y -2 z

Example 2: Use simplifying exponents rules to simplify (2a 3 b 5 c) × (5ab 6 c 2 ).

Solution: We will combine the like terms and simplify them separately. So, we have

(2a 3 b 5 c) × (5ab 6 c 2 )

= (2×5) (a 3 ×a) (b 5 ×b 6 ) (c×c 2 )

= 10 a 3+1 b 5+6 c 1+2

= 10a 4 b 11 c 3

Answer: (2a 3 b 5 c) × (5ab 6 c 2 ) = 10a 4 b 11 c 3

Simplifying Rational Exponents

Now that we have understood how to apply simplifying exponents rules, let us now learn to simplify rational exponents . We apply the rules in the same way for simplifying rational exponents as we did for whole numbers. Some of the common rules that we will use here are:

• a x/y × a m/n = a x/y + m/n
• a x/y ÷ a m/n = a x/y - m/n
• (a/b) m/n = (b/a) -m/n
• a m/n = (1/a) -m/n
• (a m/n ) x/y = a (m/n) × (x/y)

Simplifying Negative Exponents

As we discussed in the previous section, for simplifying negative exponents , we apply the laws of exponents in the same way. Some of the common rules that we use for simplifying such exponents are:

• a -m × a -n = a ( - m)+(-n)
• a - m /a - n = a - m-(-n)
• a -m = 1/a m ; (a/b) -m = (b/a) m
• (a -m ) - n = a -m×-n
• (ab) - m = a -m b -m
• (a/b) -m = a -m /b -m

Important Notes on Simplifying Exponents

• Simplifying exponents is a simple process of reducing the mathematical expressions involving exponents into a simpler form such that they cannot further be simplified.
• When we apply arithmetic operations on exponents, we use the laws of exponents for simplifying exponent expressions.
• We can apply the rules of simplifying exponents for simplifying rational and negative exponents.

☛ Related Articles:

• Irrational Exponents
• Exponents Formula
• Exponential Equations

Simplifying Exponents Examples

Example 1: Simplify (4a 3 b 6 c -3 ) ÷ (2a 4 bc 2 )

Solution: For simplifying exponent expression (4a 3 b 6 c -3 ) ÷ (2a 4 bc 2 ), we have

(4a 3 b 6 c -3 ) ÷ (2a 4 bc 2 )

= (4/2) (a 3 /a 4 ) (b 6 /b) (c -3 /c 2 )

= 2 a 3-4 b 6-1 c -3-2

= 2a -1 b 5 c -5

= 2b 5 /(ac 5 )

Answer: (4a 3 b 6 c -3 ) ÷ (2a 4 bc 2 ) = 2b 5 /(ac 5 )

Example 2: Evaluate 4 3 × 4 -1 using simplifying exponents rules.

Solution: We will use the rule a m × a n = a m+n to simplify the given expression. So, we have

= 4 3 + (-1)

Answer: 4 3 × 4 -1 = 16

Example 3: Simplify [a -1/2 / b -2/3 ] 1/2

Solution: Using the rules of simplifying rational exponents, we have

[a -1/2 / b -2/3 ] 1/2

= [a -1/2 ] 1/2 / [b -2/3 ] 1/2

= a -1/4 / b -1/3

Answer: [a -1/2 / b -2/3 ] 1/2 = a -1/4 / b -1/3

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Simplifying Exponents Questions

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FAQs on Simplifying Exponents

What is simplifying exponents in math.

Simplifying exponents is a method of simplifying the algebraic expressions involving exponents into a simpler form such that they cannot further be simplified. There are rules in algebra for simplifying exponents with different and same bases.

What are the Rules for Simplifying Exponents?

Given below is a list of rules that we for simplifying exponents in algebraic expressions:

• Product Rule: a m × a n = a m+n
• Quotient Rule: a m /a n = a m-n
• Power of a Power Rule: (a m ) n = a mn

What is Simplifying Exponents in Fractions?

When algebraic expressions involving exponents are given in fractions, we simplify them using the laws of exponents. For example, (35x 3 y 2 z) / (7xy 4 ), (5x 2 y 7 z) / (xy -1 ), etc.

How to Simplify Negative Exponents?

For simplifying expressions with negative exponents, we use the same laws of exponents as we use for whole numbers. Some of the commonly used rules are:

How to Simplify Rational Exponents?

We apply the rules in the same way for simplifying rational exponents as we did for whole numbers. Some of the common rules that we will use here are:

What is Simplifying Exponents with Different Bases?

When we multiply exponents or divide exponents with different bases, there can be two cases: i) when the exponents are the same, ii) when the exponents are different. When simplifying exponents with different bases and the same power, we follow the rule:

When we have to simplify exponents with different bases and different power, we simplify the terms separately and then apply the arithmetic operation involved.

What is the Difference Between Simplifying Exponents and Evaluating Exponents?

Simplifying exponents means reducing an expression with an exponent to a simpler form such that it cannot further be reduced. Evaluating exponents implied determining the value of an expression by substituting the value of the variable involved or by doing the calculations involved.

Exponents and Scientific Notation

Simplifying exponential expressions.

Recall that to simplify an expression means to rewrite it by combing terms or exponents; in other words, to write the expression more simply with fewer terms. The rules for exponents may be combined to simplify expressions.

Example 9: Simplifying Exponential Expressions

Simplify each expression and write the answer with positive exponents only.

• [latex]{\left(6{m}^{2}{n}^{-1}\right)}^{3}[/latex]
• [latex]{17}^{5}\cdot {17}^{-4}\cdot {17}^{-3}[/latex]
• [latex]{\left(\frac{{u}^{-1}v}{{v}^{-1}}\right)}^{2}[/latex]
• [latex]\left(-2{a}^{3}{b}^{-1}\right)\left(5{a}^{-2}{b}^{2}\right)[/latex]
• [latex]{\left({x}^{2}\sqrt{2}\right)}^{4}{\left({x}^{2}\sqrt{2}\right)}^{-4}[/latex]
• [latex]\frac{{\left(3{w}^{2}\right)}^{5}}{{\left(6{w}^{-2}\right)}^{2}}[/latex]
• [latex]\begin{array}{cccc}\hfill {\left(6{m}^{2}{n}^{-1}\right)}^{3}& =& {\left(6\right)}^{3}{\left({m}^{2}\right)}^{3}{\left({n}^{-1}\right)}^{3}\hfill & \text{The power of a product rule}\hfill \\ & =& {6}^{3}{m}^{2\cdot 3}{n}^{-1\cdot 3}\hfill & \text{The power rule}\hfill \\ & =& \text{ }216{m}^{6}{n}^{-3}\hfill & \text{Simplify}.\hfill \\ & =& \frac{216{m}^{6}}{{n}^{3}}\hfill & \text{The negative exponent rule}\hfill \end{array}[/latex]
• [latex]\begin{array}{cccc}\hfill {17}^{5}\cdot {17}^{-4}\cdot {17}^{-3}& =& {17}^{5 - 4-3}\hfill & \text{The product rule}\hfill \\ & =& {17}^{-2}\hfill & \text{Simplify}.\hfill \\ & =& \frac{1}{{17}^{2}}\text{ or }\frac{1}{289}\hfill & \text{The negative exponent rule}\hfill \end{array}[/latex]
• [latex]\begin{array}{cccc}\hfill {\left(\frac{{u}^{-1}v}{{v}^{-1}}\right)}^{2}& =& \frac{{\left({u}^{-1}v\right)}^{2}}{{\left({v}^{-1}\right)}^{2}}\hfill & \text{The power of a quotient rule}\hfill \\ & =& \frac{{u}^{-2}{v}^{2}}{{v}^{-2}}\hfill & \text{The power of a product rule}\hfill \\ & =& {u}^{-2}{v}^{2-\left(-2\right)}& \text{The quotient rule}\hfill \\ & =& {u}^{-2}{v}^{4}\hfill & \text{Simplify}.\hfill \\ & =& \frac{{v}^{4}}{{u}^{2}}\hfill & \text{The negative exponent rule}\hfill \end{array}[/latex]
• [latex]\begin{array}{cccc}\hfill \left(-2{a}^{3}{b}^{-1}\right)\left(5{a}^{-2}{b}^{2}\right)& =& -2\cdot 5\cdot {a}^{3}\cdot {a}^{-2}\cdot {b}^{-1}\cdot {b}^{2}\hfill & \text{Commutative and associative laws of multiplication}\hfill \\ & =& -10\cdot {a}^{3 - 2}\cdot {b}^{-1+2}\hfill & \text{The product rule}\hfill \\ & =& -10ab\hfill & \text{Simplify}.\hfill \end{array}[/latex]
• [latex]\begin{array}{cccc}\hfill {\left({x}^{2}\sqrt{2}\right)}^{4}{\left({x}^{2}\sqrt{2}\right)}^{-4}& =& {\left({x}^{2}\sqrt{2}\right)}^{4 - 4}\hfill & \text{The product rule}\hfill \\ & =& \text{ }{\left({x}^{2}\sqrt{2}\right)}^{0}\hfill & \text{Simplify}.\hfill \\ & =& 1\hfill & \text{The zero exponent rule}\hfill \end{array}[/latex]
• [latex]\begin{array}{cccc}\hfill \frac{{\left(3{w}^{2}\right)}^{5}}{{\left(6{w}^{-2}\right)}^{2}}& =& \frac{{\left(3\right)}^{5}\cdot {\left({w}^{2}\right)}^{5}}{{\left(6\right)}^{2}\cdot {\left({w}^{-2}\right)}^{2}}\hfill & \text{The power of a product rule}\hfill \\ & =& \frac{{3}^{5}{w}^{2\cdot 5}}{{6}^{2}{w}^{-2\cdot 2}}\hfill & \text{The power rule}\hfill \\ & =& \frac{243{w}^{10}}{36{w}^{-4}}\hfill & \text{Simplify}.\hfill \\ & =& \frac{27{w}^{10-\left(-4\right)}}{4}\hfill & \text{The quotient rule and reduce fraction}\hfill \\ & =& \frac{27{w}^{14}}{4}\hfill & \text{Simplify}.\hfill \end{array}[/latex]

a. [latex]{\left(2u{v}^{-2}\right)}^{-3}[/latex] b. [latex]{x}^{8}\cdot {x}^{-12}\cdot x[/latex] c. [latex]{\left(\frac{{e}^{2}{f}^{-3}}{{f}^{-1}}\right)}^{2}[/latex] d. [latex]\left(9{r}^{-5}{s}^{3}\right)\left(3{r}^{6}{s}^{-4}\right)[/latex] e. [latex]{\left(\frac{4}{9}t{w}^{-2}\right)}^{-3}{\left(\frac{4}{9}t{w}^{-2}\right)}^{3}[/latex] f. [latex]\frac{{\left(2{h}^{2}k\right)}^{4}}{{\left(7{h}^{-1}{k}^{2}\right)}^{2}}[/latex]

• College Algebra. Authored by : OpenStax College Algebra. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected]:1/Preface . License : CC BY: Attribution

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Unit 11: Exponents & radicals

About this unit.

Let's review exponent rules and level up what we know about roots. The square root is nice, but let's learn about higher-order roots like the cube root (or 3rd root).

Exponent properties review

• Multiplying & dividing powers (integer exponents) (Opens a modal)
• Powers of products & quotients (integer exponents) (Opens a modal)
• Multiply & divide powers (integer exponents) Get 5 of 7 questions to level up!
• Powers of products & quotients (integer exponents) Get 3 of 4 questions to level up!
• Properties of exponents challenge (integer exponents) Get 3 of 4 questions to level up!
• Intro to square roots (Opens a modal)
• Understanding square roots (Opens a modal)
• Square root of decimal (Opens a modal)
• Intro to cube roots (Opens a modal)
• 5th roots (Opens a modal)
• Higher order roots (Opens a modal)
• Square roots Get 5 of 7 questions to level up!
• Roots of decimals & fractions Get 3 of 4 questions to level up!
• Cube roots Get 5 of 7 questions to level up!

Simplifying square roots

• Simplifying square roots (Opens a modal)
• Simplifying square roots (variables) (Opens a modal)
• Simplifying square-root expressions (Opens a modal)
• Simplifying square roots review (Opens a modal)
• Exponents & radicals: FAQ (Opens a modal)
• Simplify square roots Get 3 of 4 questions to level up!
• Simplify square roots (variables) Get 3 of 4 questions to level up!
• Simplify square-root expressions Get 3 of 4 questions to level up!

Common Core State Standards Initiative

High School: Algebra

Standards in this domain:, ccss.math.content.hsa.introduction introduction, expressions..

An expression is a record of a computation with numbers, symbols that represent numbers, arithmetic operations, exponentiation, and, at more advanced levels, the operation of evaluating a function. Conventions about the use of parentheses and the order of operations assure that each expression is unambiguous. Creating an expression that describes a computation involving a general quantity requires the ability to express the computation in general terms, abstracting from specific instances.

Reading an expression with comprehension involves analysis of its underlying structure. This may suggest a different but equivalent way of writing the expression that exhibits some different aspect of its meaning. For example, p + 0.05 p can be interpreted as the addition of a 5% tax to a price p . Rewriting p + 0.05 p as 1.05 p shows that adding a tax is the same as multiplying the price by a constant factor.

Algebraic manipulations are governed by the properties of operations and exponents, and the conventions of algebraic notation. At times, an expression is the result of applying operations to simpler expressions. For example, p + 0.05 p is the sum of the simpler expressions p and 0.05 p . Viewing an expression as the result of operation on simpler expressions can sometimes clarify its underlying structure.

A spreadsheet or a computer algebra system (CAS) can be used to experiment with algebraic expressions, perform complicated algebraic manipulations, and understand how algebraic manipulations behave.

Equations and inequalities.

An equation is a statement of equality between two expressions, often viewed as a question asking for which values of the variables the expressions on either side are in fact equal. These values are the solutions to the equation. An identity, in contrast, is true for all values of the variables; identities are often developed by rewriting an expression in an equivalent form.

The solutions of an equation in one variable form a set of numbers; the solutions of an equation in two variables form a set of ordered pairs of numbers, which can be plotted in the coordinate plane. Two or more equations and/or inequalities form a system. A solution for such a system must satisfy every equation and inequality in the system.

An equation can often be solved by successively deducing from it one or more simpler equations. For example, one can add the same constant to both sides without changing the solutions, but squaring both sides might lead to extraneous solutions. Strategic competence in solving includes looking ahead for productive manipulations and anticipating the nature and number of solutions.

Some equations have no solutions in a given number system, but have a solution in a larger system. For example, the solution of x + 1 = 0 is an integer, not a whole number; the solution of 2 x + 1 = 0 is a rational number, not an integer; the solutions of x 2 - 2 = 0 are real numbers, not rational numbers; and the solutions of x 2 + 2 = 0 are complex numbers, not real numbers.

The same solution techniques used to solve equations can be used to rearrange formulas. For example, the formula for the area of a trapezoid, A = (( b 1 + b 2 )/2) h , can be solved for h using the same deductive process. Inequalities can be solved by reasoning about the properties of inequality. Many, but not all, of the properties of equality continue to hold for inequalities and can be useful in solving them.

Connections to Functions and Modeling. Expressions can define functions, and equivalent expressions define the same function. Asking when two functions have the same value for the same input leads to an equation; graphing the two functions allows for finding approximate solutions of the equation. Converting a verbal description to an equation, inequality, or system of these is an essential skill in modeling.

Algebra Overview

• Seeing Structure in Expressions
• Interpret the structure of expressions
• Write expressions in equivalent forms to solve problems

Arithmetic with Polynomials and Rational Functions

• Perform arithmetic operations on polynomials
• Understand the relationship between zeros and factors of polynomials
• Use polynomial identities to solve problems
• Rewrite rational functions

Creating Equations

• Create equations that describe numbers or relationships

Reasoning with Equations and Inequalities

• Understand solving equations as a process of reasoning and explain the reasoning
• Solve equations and inequalities in one variable
• Solve systems of equations
• Represent and solve equations and inequalities graphically

Mathematical Practices

• Make sense of problems and persevere in solving them.
• Reason abstractly and quantitatively.
• Construct viable arguments and critique the reasoning of others.
• Model with mathematics.
• Use appropriate tools strategically.
• Attend to precision.
• Look for and make use of structure.
• Look for and express regularity in repeated reasoning.

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• Pre Algebra
• One-Step Addition
• One-Step Subtraction
• One-Step Multiplication
• One-Step Division
• One-Step Decimals
• Two-Step Integers
• Two-Step Add/Subtract
• Two-Step Multiply/Divide
• Two-Step Fractions
• Two-Step Decimals
• Multi-Step Integers
• Multi-Step with Parentheses
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• Solve by Factoring
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• Absolute Value
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• Powers of i
• Complex Form
• Partial Fractions
• Is Polynomial
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• Leading Term
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• Complete the Square
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• Rationalize Denominator
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• Identify Type
• Convergence
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• Age Problems
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