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Exponent Rules

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Exponent Rules
Law of Exponents for Real Numbers
If mmm and nnn are any real number, then:

1.am×an=am+n2.aman=am−n(a≠0)3.(am)n=amn4.(ab)m=ambm5.(ab)m=ambm(b≠0)6.a0=1(a≠0)7.am/n=nam8.a−m=1am(a≠0)\begin{array}{lccl}
1.&a^m\times{}a^n&=&a^{m+n}\\\\
2.&\dfrac{a^m}{a^n}&=&a^{m-n}&&(a\neq0)\\\\
3.&(a^m)^n&=&a^{mn}\\\\
4.&(ab)^{m}&=&a^mb^m\\\\
5.&\Bigg(\dfrac{a}{b}\Bigg)^m&=&\dfrac{a^m}{b^m}&&(b\neq0)\\\\
6.&a^0&=&1&&(a\neq0)\\\\
7.&a^{m/n}&=&^n\sqrt{a^m}\\\\
8.&a^{-m}&=&\dfrac{1}{a^m}&&(a\neq0)
\end{array}1.2.3.4.5.6.7.8.​am×ananam​(am)n(ab)m(ba​)ma0am/na−m​========​am+nam−namnambmbmam​1nam​am1​​​(a=0)(b=0)(a=0)(a=0)​


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Example

Simplify −28x2y−54a−7b3\dfrac{-28x^2y^{-5}}{4a^{-7}b^3}4a−7b3−28x2y−5​. Make sure all exponents are positive. 

−28x2y−54x−7y3=−7x(2−(−7))y(−5−3)=−7x5y−8=−7x5y8\begin{array}{rcl}
\dfrac{-28x^2y^{-5}}{4x^{-7}y^3}&=&-7x^{(2-(-7))}y^{(-5-3)}\\\\
&=&-7x^{5}y^{-8}\\\\
&=&-\dfrac{7x^5}{y^8}
\end{array}4x−7y3−28x2y−5​​===​−7x(2−(−7))y(−5−3)−7x5y−8−y87x5​​

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Example: Exponent Rules
Simplify, but do not evaluate:
(82/3)(163/2)(8^{2/3})(16^{3/2})(82/3)(163/2)
(9a3b−7c0)(18a−5b2)−81a−5b−1c4\dfrac{(9a^3b^{-7}c^0)(18a^{-5}b^2)}{-81a^{-5}b^{-1}c^4}−81a−5b−1c4(9a3b−7c0)(18a−5b2)​
41/2+(12)44^{1/2}+\Bigg(\dfrac{1}{2}\Bigg)^441/2+(21​)4

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Part a. 

(82/3)(163/2)=(23)23(24)32=(22)(26)=28\begin{array}{rcl}
(8^{2/3})(16^{3/2})&=&(2^{3})^{\frac{2}{3}}(2^{4})^{\frac{3}{2}}\\\\
&=&(2^2)(2^6)\\\\
&=&\boxed{2^8}
\end{array}(82/3)(163/2)​===​(23)32​(24)23​(22)(26)28​​

Part b.

(9a3b−7c0)(18a−5b2)−81a−5b−1c4=((32a3b−7c0)(2×32a−5b2)−34a−5b−1c4)=−2×3(2+2−4)a(3−5−(−5))b(−7+2−(−1))c(0−4)=−2a3b−4c−4=−2a3b4c4\begin{array}{rcl}
\dfrac{(9a^3b^{-7}c^0)(18a^{-5}b^2)}{-81a^{-5}b^{-1}c^4}&=&\Bigg(\dfrac{(3^2a^3b^{-7}c^0)(2 \times3^2a^{-5}b^2)}{-3^4a^{-5}b^{-1}c^4}\Bigg)\\\\
&=&-2\times3^{(2+2-4)}a^{(3-5-(-5))}b^{(-7+2-(-1))}c^{(0-4)}\\\\
&=&-2a^{3}b^{-4}c^{-4}\\\\
&=&\boxed{-\dfrac{2a^3}{b^4c^4}}
\end{array}−81a−5b−1c4(9a3b−7c0)(18a−5b2)​​====​(−34a−5b−1c4(32a3b−7c0)(2×32a−5b2)​)−2×3(2+2−4)a(3−5−(−5))b(−7+2−(−1))c(0−4)−2a3b−4c−4−b4c42a3​​​

Part c. 

41/2+(12)4=(22)12+(2−1)4=2+2−4=2+124Common Denominator is 24=2⋅(2424)+124=2524+124=25+124\begin{array}{rcl}
4^{1/2}+\Bigg(\dfrac{1}{2}\Bigg)^4&=&(2^2)^{\frac{1}{2}}+(2^{-1})^4\\\\
&=&2+2^{-4}\\\\
&=&2+\dfrac{1}{2^4}&&\colorTwo{\footnotesize{\text{Common Denominator is}~2^4}}\\\\
&=&2\cdot\Bigg(\dfrac{2^4}{2^4}\Bigg)+\dfrac{1}{2^4}\\\\
&=&\dfrac{2^5}{2^4}+\dfrac{1}{2^4}\\\\
&=&\boxed{\dfrac{2^5+1}{2^4}}
\end{array}41/2+(21​)4​======​(22)21​+(2−1)42+2−42+241​2⋅(2424​)+241​2425​+241​2425+1​​​Common Denominator is 24

Practice: Exponent Rules
Evaluate (49)−32÷(1625)−12\Bigg(\dfrac{4}{9}\Bigg)^{-\frac{3}{2}}\div{}\Bigg(\dfrac{16}{25}\Bigg)^{-\frac{1}{2}}(94​)−23​÷(2516​)−21​. Leave answer in exact form. 

Answer

I don't know

Practice: Exponent Rules
Simplify ((4a−3/4b−1/2)(64a−1/2b−3/2)(256b1/4)(−16a−3/2b3/4))\Bigg(\dfrac{(4a^{-3/4}b^{-1/2})(64a^{-1/2}b^{-3/2})}{(256b^{1/4})(-16a^{-3/2}b^{3/4})}\Bigg)((256b1/4)(−16a−3/2b3/4)(4a−3/4b−1/2)(64a−1/2b−3/2)​). Make all exponents positive.

A) −a1424b3-\dfrac{a^{\frac{1}{4}}}{2^4b^3}−24b3a41​​   

B) −24a14b3-2^4a^{\frac{1}{4}}b^3−24a41​b3 

C) a5424\dfrac{a^{\frac{5}{4}}}{2^4}24a45​​ 

D)  24a5/4b3\dfrac{2^4a^{5/4}}{b^3}b324a5/4​ 

I don't know

Practice: Exponent Rules
If x=18x=\dfrac{1}{8}x=81​ and y=4y=4y=4, simplify the expression (x−2y1/2)(x3y3/2)−1(x^{-2}y^{1/2})(x^3y^{3/2})^{-1}(x−2y1/2)(x3y3/2)−1. Leave all exponents positive. 

Answer

I don't know

Practice: Exponent Rules
True or false: 64x+64x+64x+64x=44x+164^x+64^x+64^x+64^x=4^{4x+1}64x+64x+64x+64x=44x+1

Answer

I don't know

Extra Practice

Simplifying Exponential Expressions

Simplify (72ab(2a+b)32a−3b)\Bigg(\dfrac{72^{ab}(2^{a+b})}{3^{2a-3b}}\Bigg)(32a−3b72ab(2a+b)​) 

Exponent Rules

Practice: Exponent Rules
True or false: 27x+27x+27x=33x+627^x+27^x+27^x=3^{3x+6}27x+27x+27x=33x+6

Exponent Rules

Practice: Exponent Rules
True or false: 1252x+1252x+1252x+1252x+1252x=56x+1125^{2x}+125^{2x}+125^{2x}+125^{2x}+125^{2x}=5^{6x+1}1252x+1252x+1252x+1252x+1252x=56x+1

Exponent Rules

Practice: Exponent Rules
If x=19x=\dfrac{1}{9}x=91​ and y=27y=27y=27, simplify the expression (x−1/2y1/3)−1(x2y2/3)(x^{-1/2}y^{1/3})^{-1}(x^2y^{2/3})(x−1/2y1/3)−1(x2y2/3). Leave all exponents positive. 

Exponent Rules

Practice: Exponent Rules
Simplify ((9x1/5y−2/7)(6x−2/5y1/7)(81x4/3)(−36x4/5y3/2))\Bigg(\dfrac{(9x^{1/5}y^{-2/7})(6x^{-2/5}y^{1/7})}{(81x^{4/3})(-36x^{4/5}y^{3/2})}\Bigg)((81x4/3)(−36x4/5y3/2)(9x1/5y−2/7)(6x−2/5y1/7)​). Make all exponents positive.

Exponent Rules

Practice: Exponent Rules
Simplify ((3a−2/3b−4/5)(25a4/3b−1/2)(81a5/6b4/3)(5a5/2b−1/5))\Bigg(\dfrac{(3a^{-2/3}b^{-4/5})(25a^{4/3}b^{-1/2})}{(81a^{5/6}b^{4/3})(5a^{5/2}b^{-1/5})}\Bigg)((81a5/6b4/3)(5a5/2b−1/5)(3a−2/3b−4/5)(25a4/3b−1/2)​). Make all exponents positive.

Exponent Rules

Practice: Exponent Rules
Evaluate (343125)−13×(648)23\Bigg(\dfrac{343}{125}\Bigg)^{-\frac{1}{3}}\times{}\Bigg(\dfrac{64}{8}\Bigg)^{\frac{2}{3}}(125343​)−31​×(864​)32​. Leave answer in exact form. 

Exponent Rules

Practice: Exponent Rules
Evaluate (649)−52÷(48416)12\Bigg(\dfrac{64}{9}\Bigg)^{-\frac{5}{2}}\div{}\Bigg(\dfrac{484}{16}\Bigg)^{\frac{1}{2}}(964​)−25​÷(16484​)21​. Leave answer in exact form. 