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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=amn(a0)3.(am)n=amn4.(ab)m=ambm5.(ab)m=ambm(b0)6.a0=1(a0)7.am/n=nam8.am=1am(a0)\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 28x2y54a7b3\dfrac{-28x^2y^{-5}}{4a^{-7}b^3}. Make sure all exponents are positive.


28x2y54x7y3=7x(2(7))y(53)=7x5y8=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. (9a3b7c0)(18a5b2)81a5b1c4\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.

(9a3b7c0)(18a5b2)81a5b1c4=((32a3b7c0)(2×32a5b2)34a5b1c4)=2×3(2+24)a(35(5))b(7+2(1))c(04)=2a3b4c4=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+(21)4=2+24=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 ((4a3/4b1/2)(64a1/2b3/2)(256b1/4)(16a3/2b3/4))\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 (x2y1/2)(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