Question

division properties of exponents
simplify each expression. assume that no denominator equals zero.

1. $\frac{a^{7}b^{8}}{ab^{5}}$
2. $\frac{c^{4}d^{5}}{c^{4}d^{2}}$
3. $\frac{20d^{3}f^{2}g^{8}}{5d^{2}fg^{3}}$
4. $\frac{-18h^{6}}{2h^{3}}$
5. $left(\frac{2j^{2}k^{5}}{3m}

ight)^{3}$

1. $left(\frac{n^{2}p}{5q^{4}}

ight)^{2}$

1. $left(\frac{4ru^{2}}{5t^{4}}

ight)^{3}$

1. $left(\frac{u^{2}v^{6}}{2w^{5}}

ight)^{4}$

Explanation:

Step1: Split into like terms

$\frac{a^7}{a} \cdot \frac{b^8}{b^5}$

Step2: Apply exponent rule $\frac{x^m}{x^n}=x^{m-n}$

$a^{7-1} \cdot b^{8-5} = a^6b^3$

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Step1: Split into like terms

$\frac{c^4}{c^4} \cdot \frac{d^5}{d^2}$

Step2: Apply exponent rule $\frac{x^m}{x^n}=x^{m-n}$

$c^{4-4} \cdot d^{5-2} = 1 \cdot d^3 = d^3$

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Step1: Split into coefficients/like terms

$\frac{20}{5} \cdot \frac{d^3}{d^2} \cdot \frac{f^2}{f} \cdot \frac{g^8}{g^3}$

Step2: Simplify each part

$4 \cdot d^{3-2} \cdot f^{2-1} \cdot g^{8-3} = 4dfg^5$

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Step1: Split into coefficients/like terms

$\frac{-18}{2} \cdot \frac{h^6}{h^3}$

Step2: Simplify each part

$-9 \cdot h^{6-3} = -9h^3$

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Step1: Apply power rule $(xy)^n=x^ny^n$

$\frac{(2j^2k^5)^3}{(3m)^3} = \frac{2^3 \cdot (j^2)^3 \cdot (k^5)^3}{3^3 \cdot m^3}$

Step2: Apply power rule $(x^m)^n=x^{mn}$

$\frac{8 \cdot j^{6} \cdot k^{15}}{27m^3} = \frac{8j^6k^{15}}{27m^3}$

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Step1: Apply power rule $(xy)^n=x^ny^n$

$\frac{(n^2p)^2}{(5q^4)^2} = \frac{(n^2)^2 \cdot p^2}{5^2 \cdot (q^4)^2}$

Step2: Apply power rule $(x^m)^n=x^{mn}$

$\frac{n^{4} \cdot p^2}{25 \cdot q^{8}} = \frac{n^4p^2}{25q^8}$

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Step1: Apply power rule $(xy)^n=x^ny^n$

$\frac{(4ru^2)^3}{(5t^4)^3} = \frac{4^3 \cdot r^3 \cdot (u^2)^3}{5^3 \cdot (t^4)^3}$

Step2: Apply power rule $(x^m)^n=x^{mn}$

$\frac{64 \cdot r^3 \cdot u^{6}}{125 \cdot t^{12}} = \frac{64r^3u^6}{125t^{12}}$

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Step1: Apply power rule $(xy)^n=x^ny^n$

$\frac{(u^2v^6)^4}{(2w^5)^4} = \frac{(u^2)^4 \cdot (v^6)^4}{2^4 \cdot (w^5)^4}$

Step2: Apply power rule $(x^m)^n=x^{mn}$

$\frac{u^{8} \cdot v^{24}}{16 \cdot w^{20}} = \frac{u^8v^{24}}{16w^{20}}$

Answer:

1. $a^6b^3$
2. $d^3$
3. $4dfg^5$
4. $-9h^3$
5. $\frac{8j^6k^{15}}{27m^3}$
6. $\frac{n^4p^2}{25q^8}$
7. $\frac{64r^3u^6}{125t^{12}}$
8. $\frac{u^8v^{24}}{16w^{20}}$