Question

simplify. write the answers with all variables in the denominator.

1. $\frac{k^{5}m^{2}}{k^{7}m^{-5}}$
2. $\frac{a^{2}bc^{-2}c^{5}}{a^{2}b^{-3}a^{2}c^{3}}$
3. expand by using the distributive property. write the answer with all variables in the denominator.

$\frac{x^{-2}p^{0}}{y^{4}}\left(\frac{x^{2}}{p^{4}} - x^{4}p^{6}\
ight)$

1. simplify by adding like terms. write the answer with all variables in the numerator.

$xy^{2} - \frac{3xy}{y^{-1}} - \frac{2x^{0}y^{2}}{x^{-1}} - \frac{4x^{2}}{y^{2}} + 2x^{2}y^{-2}$

Explanation:

Problem 21

Step1: Subtract exponents of same base

$\frac{k^5}{k^7} = k^{5-7}$, $\frac{m^2}{m^{-5}} = m^{2-(-5)}$

Step2: Simplify exponents

$k^{-2}m^{7} = \frac{m^7}{k^2}$

Step1: Split into product of fractions

$\frac{a^2}{a^2} \cdot \frac{b}{b^{-3}} \cdot \frac{c^{-2}}{a^2} \cdot \frac{c^5}{c^3}$

Step2: Simplify each fraction

$1 \cdot b^{1-(-3)} \cdot a^{0-2} \cdot c^{-2+5-3}$

Step3: Calculate final exponents

$b^4a^{-2}c^{0} = \frac{b^4}{a^2}$

Step1: Apply distributive property

$\frac{x^{-2}p^0}{y^4} \cdot \frac{x^2}{p^4} - \frac{x^{-2}p^0}{y^4} \cdot x^4p^6$

Step2: Simplify $p^0$ (=$1$) and multiply terms

$\frac{x^{-2+2}}{y^4p^4} - \frac{x^{-2+4}p^6}{y^4}$

Step3: Simplify exponents

$\frac{x^0}{y^4p^4} - \frac{x^2p^6}{y^4} = \frac{1}{y^4p^4} - \frac{x^2p^6}{y^4}$

Step4: Combine into single fraction

$\frac{1 - x^2p^{10}}{y^4p^4}$

Answer:

$\frac{m^7}{k^2}$

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Problem 22