2.7: The Power Rules for Exponents
Overview
- The Power Rule for Powers
- The Power Rule for Products
- The Power Rule for quotients
The Power Rule for Powers
The following examples suggest a rule for raising a power to a power:
\((a^2)^3 = a^2 \cdot a^2 \cdot a^2\)
Using the product rule we get:
\((a^2)^3 = a^{2+2+2}\)
\((a^2)^3 = a^{3 \cdot 2}\)
\((a^2)^3 = a^6\)
\((x^9)^4 = x^9 \cdot x^9 \cdot x^9 \cdot x^9\)
\((x^9)^4 = x^{9+9+9+9}\)
\((x^9)^4 = x^{4 \cdot 9}\)
\((x^9)^4 = x^{36}\)
If \(x\) is a real number and \(n\) and \(m\) are natural numbers,
\((x^n)^m = x^{n \cdot m}\)
To raise a power to a power, multiply the exponents.
Sample Set A
Simplify each expression using the power rule for powers. All exponents are natural numbers.
\((x^3)^4 = x^{3 \cdot 4} = x^{12}\)
\((y^5)^3 = y^{5 \cdot 3} = y^{15}\)
\((d^{20})^6 = d^{20 \cdot 6} = d^{120}\)
\((x^□)^△ = x^{□△}\)
Although we don’t know exactly what number □△ is, the notation □△ indicates the multiplication.
Practice Set A
Simplify each expression using the power rule for powers.
\((x^5)^4\)
- Answer
-
\(x^{20}\)
\((y^7)^7\)
- Answer
-
\(y^{49}\)
The Power Rule for Products
The following examples suggest a rule for raising a product to a power:
\(
\begin{aligned}
&(a b)^{3}=a b \cdot a b \cdot a b \text { Use the commutative property of multiplication. }\\
&\begin{array}{l}
=a a a b b b \\
=a^{3} b^{3}
\end{array}
\end{aligned}
\)
\(
\begin{aligned}
(x y)^{5} &=x y \cdot x y \cdot x y \cdot x y \cdot x y \\
&=x x x x x \cdot \text { yyyyy } \\
&=x^{5} y^{5}
\end{aligned}
\)
\(
\begin{aligned}
(4 x y \mathrm{z})^{2} &=4 x y z \cdot 4 x y z \\
&=4 \cdot 4 \cdot x x \cdot y y \cdot z z \\
&=16 x^{2} y^{2} z^{2}
\end{aligned}
\)
If \(x\) and \(y\) are real numbers are \(n\) is a natural number,
\((xy)^n = x^ny^n\)
To raise a product to a power, apply the exponent rule to each and every factor
Sample Set B
Make use of either or both the power rule for products and power rule for powers to simplify each expression.
\((ab)^7 = a^7b^7\)
\((axy)^4 = a^4x^4y^4\)
\((3ab)^2 = 3^2a^2b^2 = 9a^2b^2\)
Don't forget to apply the exponent to the 3!
\((2st)^5 = 2^5s^5t^5 = 32s^5t^5\)
\((ab^3)^2 = a^2(b^3)^2 = a^2b^6\)
We used two rules here. First, the power rule for products. Second, the power rule for powers.
\((7a^4b^2c^8)^2 = 7^2(a^4)^2(b^2)^2(c^8)^2 = 49a^8b^4c^{16}\)
If \(6a^3c^7 \not = 0\), then \((6a^3c^7)^0 = 1\). Recall that \(x^0 = 1\) for \(x \not = 0\)
\([2(x+1)^4]^6 = 2^6(x+1)^{24} = 64(x+1)^{24}\)
Practice Set B
Make use of either or both the power rule for products and the power rule for powers to simplify each expression.
\((ax)^4\)
- Answer
-
\(a^4x^4\)
\((3bxy)^2\)
- Answer
-
\(9b^2x^2y^2\)
\([4t(s-5)]^3\)
- Answer
-
\(64t^3(s-5)^3\)
\((9x^3y^5)^2\)
- Answer
-
\(81x^6y^{10}\)
\((1a^5b^8c^3d)^6\)
- Answer
-
\(a^{30}b^{48}c^{18}d^6\)
\([(a+8)(a+5)]^4\)
- Answer
-
\((a+8)^4(a+5)^4\)
\([(12c^4u^3(w-3)^2]^5\)
- Answer
-
\(12^5c^{20}u^{15}(w-3)^{10}\)
\([10t^4y^7j^3d^2v^6n^4g^8(2-k)^{17}]^4\)
- Answer
-
\(10^4t^{16}y^{28}j^{12}d^8v^{24}n^{16}g^{32}(2-k)^{68}\)
\((x^3x^5y^2y^6)^9\)
- Answer
-
\((x^8y^8)^9 = x^{72}y^{72}\)
\((10^6 \cdot 10^{12} \cdot 10^5)^{10}\)
- Answer
-
\(10^{230}\)
The Power Rule for Quotients
The following example suggests a rule for raising a quotient to a power.
\((\dfrac{a}{b})^3 = \dfrac{a}{b} \cdot \dfrac{a}{b} \cdot \dfrac{a}{b} = \dfrac{a \cdot a \cdot a}{b \cdot b \cdot b} = \dfrac{a^3}{b^3}\)
If \(x\) and \(y\) are real numbers and \(n\) is a natural number,
\((\dfrac{x}{y})^n = \dfrac{x^n}{y^n}, y \not = 0\)
To raise a quotient to a power, distribute the exponent to both the numerator and denominator.
Sample Set C
Make use of the power rule for quotients, the power rule for products, the power rule for powers, or a combination of these rules to simplify each expression. All exponents are natural numbers.
\((\dfrac{x}{y})^6 = \dfrac{x^6}{y^6}\)
\((\dfrac{a}{c})^2 = \dfrac{a^2}{c^2}\)
\((\dfrac{2x}{b})^4 = \dfrac{(2x)^4}{b^4} = \dfrac{2^4x^4}{b^4} = \dfrac{16x^4}{b^4}\)
\((\dfrac{a^3}{b^5})^7 = \dfrac{(a^3)^7}{(b^5)^7} = \dfrac{a^21}{b^35}\)
\((\dfrac{3c^4r^2}{2^3g^5})^3 = \dfrac{3^3c^12r^6}{2^9g^{15}} = \dfrac{27c^{12}r^6}{2^9g^{15}} \text {or} \dfrac{27c^{12}r^6}{512g^15}\)
\([\dfrac{(a-2)}{(a+7)}]^4 = \dfrac{(a-2)^4}{(a+7)^4}\)
\([\dfrac{6x(4-x)^4}{2a(y-4)^6}]^2 = \dfrac{6^2x^2(4-x)^8}{2^2a^2(y-4)^{12}} = \dfrac{36x^2(4-x)^8}{4a^2(y-4)^{12}} = \dfrac{9x^2(4-x)^8}{a^2(y-4)^{12}}\)
\(
\left(\dfrac{a^{3} b^{5}}{a^{2} b}\right)^{3}=\left(a^{3-2} b^{5-1}\right)^{3}
\)
We can simplify within the parentheses. We have a rule that tells us to proceed this way
\(
=\left(a b^{4}\right)^{3}
=a^{3} b^{12}
\left(\dfrac{a^{3} b^{5}}{a^{2} b}\right)^{3}=\dfrac{a^{9} b^{15}}{a^{6} b^{3}}=a^{9-6} b^{15-3}=a^{3} b^{12}
\)
We could have actually used the power rule for quotients first.
Distribute the exponent, then simplify using the other rules.
It is probably better, for the sake of consistency, to work inside the parentheses first.
\((\dfrac{a^rb^s}{s^t})^w = \dfrac{a^{rw}b^{sw}}{c^{tw}}\)
Practice Set C
Make use of the power rule for quotients, the power rule for products, the power rule for powers, or a combination of these rules to simplify each expression.
\((\dfrac{a}{c})^5\)
- Answer
-
\(\dfrac{a^5}{c^5}\)
\((\dfrac{2x}{3y})^3\)
- Answer
-
\(\dfrac{8x^3}{27y^3}\)
\((\dfrac{x^2y^4z^7}{a^5b})^9\)
- Answer
-
\(\dfrac{x^{18}y^{36}z^{63}}{a^{45}b^9}\)
\([\dfrac{2a^4(b-1)}{3b^3(c+6)}]^4\)
- Answer
-
\(\dfrac{16a^{16}(b-1)^4}{81b^{12}(c+6)^4}\)
\((\dfrac{8a^3b^2c^6}{4a^2b})^3\)
- Answer
-
\(8a^3b^3c^{18}\)
\([\dfrac{(9+w)^2}{(3+w)^5}]^{10}\)
- Answer
-
\(\dfrac{(9+w)^{20}}{(3+w)^{50}}\)
\([\dfrac{5x^4(y+1)}{5x^4(y+1)}]^6\)
- Answer
-
\(1\), if \(x^4(y+1) \not = 0\)
\((\dfrac{16x^3v^4c^7}{12x^2vc^6})^0\)
- Answer
-
\(1\), if \(x^2vc^6 \not = 0\)
Exercises
Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers.
\((ac)^5\)
- Answer
-
\(a^5c^5\)
\((nm)^7\)
\((2a)^3\)
- Answer
-
\(8a^3\)
\((2a)^5\)
\((3xy)^4\)
- Answer
-
\(81x^4y^4\)
\((2xy)^5\)
\((3ab)^4\)
- Answer
-
\(81a^4b^4\)
\((6mn)^2\)
\((7y^3)^2\)
- Answer
-
\(49y^6\)
\((3m^3)^4\)
\((5x^6)^3\)
\((10a^2b)^2\)
- Answer
-
\(100a^4b^2\)
\((8x^2y^3)^2\)
\((x^2y^3z^5)^4\)
- Answer
-
\(x^8y^{12}z^{20}\)
\((2a^5b^{11})^0\)
\((x^3y^2z^4)^5\)
- Answer
-
\(x^{15}y^{10}z^{20}\)
\((m^6n^2p^5)^5\)
\((a^4b^7c^6d^8)^8\)
- Answer
-
\(a^{32}b^{56}c^{48}d^{64}\)
\((x^2y^3z^9w^7)^3\)
\((9xy^3)^0\)
- Answer
-
\(1\)
\((\dfrac{1}{2}f^2r^6s^5)^4\)
\((\dfrac{1}{8}c^{10}d^8e^4f^9)^2\)
- Answer
-
\(\dfrac{1}{64}c^{20}d^{16}e^{8}f^{18}\)
\((\dfrac{3}{5}a^3b^5c^{10})^3\)
\((xy)^4(x^2y^4)\)
- Answer
-
\(x^6y^8\)
\((2a^2)^4(3a^5)^2\)
\((a^2b^3)^3(a^3b^3)^4\)
- Answer
-
\(a^{18}b^{21}\)
\((h^3k^5)^2(h^2k^4)^3\)
\((x^4y^3z)^4(x^5yz^2)^2\)
- Answer
-
\(x^{26}y^{14}z^8\)
\((ab^3c^2)^5(a^2b^2c)^2\)
\(\dfrac{(6a^2b^8)^2}{(3ab^5)^2}\)
- Answer
-
\(4a^2b^6\)
\(\dfrac{(a^3b^4)^5}{(a^4b^4)^3}\)
\(\dfrac{(x^6y^5)^3}{(x^2y^3)^5}\)
- Answer
-
\(x^8\)
\(\dfrac{(a^8b^{10})^3}{(a^7b^5)^3}\)
\(\dfrac{(m^5n^6p^4)^4}{(m^4n^5p)^4}\)
- Answer
-
\(m^4n^4p^{12}\)
\(\dfrac{(x^8y^3z^2)^5}{(x^6yz)^6}\)
\(\dfrac{(10x^4y^5z^{11})^3}{(xy^2)^4}\)
- Answer
-
\(100x^8y^7z^{33}\)
\(\dfrac{(9a^4b^5)(2b^2c)}{(3a^3b)(6bc)}\)
\(\dfrac{(2x^3y^3)^4(5x^6y^8)^2}{(4x^5y^3)^2}\)
- Answer
-
\(25x^{14}y^{22}\)
\((\dfrac{3x}{5y})^2\)
\((\dfrac{3ab}{4xy})^3\)
- Answer
-
\(\dfrac{27a^3b^3}{64x^3y^3}\)
\((\dfrac{x^2y^2}{2z^3})^5\)
\((\dfrac{3a^2b^3}{c^4})^3\)
- Answer
-
\(\dfrac{27a^6b^9}{c^{12}}\)
\((\dfrac{4^2a^3b^7}{b^5c^4})^2\)
\([\dfrac{x^2(y-1)^3}{(x+6)}]^4\)
- Answer
-
\(\dfrac{x^8(y-1)^{12}}{(x+6)^4}\)
\((x^nt^{2m})^4\)
\(\dfrac{(x^{n+2})^3}{x^{2n}}\)
- Answer
-
\(x^{n+6}\)
\((xy)^△\)
\(\dfrac{4^3a^Δa^□}{4a^∇}\)
\((\dfrac{4x^Δ}{2y^∇})^□\)
- Answer
-
\(\dfrac{2^□x^{Δ□}}{y^{∇□}}\)