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Wize High School Grade 12 Pre-Calculus Textbook > Exponential Functions

Exponent Rules

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Exponent Rules

Law of Exponents for Real Numbers

If mm and nn 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}


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Example

Simplify −28x2y−54a−7b3\dfrac{-28x^2y^{-5}}{4a^{-7}b^3}. 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}
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Example: Exponent Rules

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

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}

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}

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}}. Leave answer in exact form.

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). Make all exponents positive.

Practice: Exponent Rules

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

Practice: Exponent Rules

True or false: 64x+64x+64x+64x=44x+164^x+64^x+64^x+64^x=4^{4x+1}
Extra Practice
Simplifying Exponential Expressions
Simplify (72ab(2a+b)32a−3b)\Bigg(\dfrac{72^{ab}(2^{a+b})}{3^{2a-3b}}\Bigg)
Exponent Rules
Practice: Exponent Rules
True or false: 27x+27x+27x=33x+627^x+27^x+27^x=3^{3x+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}
Exponent Rules
Practice: Exponent Rules
If x=19x=\dfrac{1}{9} and y=27y=27, simplify the expression (x−1/2y1/3)−1(x2y2/3)(x^{-1/2}y^{1/3})^{-1}(x^2y^{2/3}). 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). 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). 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}}. 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}}. Leave answer in exact form.