Question

use the properties of exponents to simplify each expression. write all answers with positive exponents only. (assume variables are nonzero.)\\(\frac{(a^{-2})^{2}(a^{2})^{-4}}{a^{10}} = \square\\)\\(\frac{4a^{5}b^{5}}{20a^{4}b^{-2}} = \square\\)

Explanation:

Step1: Simplify numerator exponents

Use power rule: $(x^m)^n=x^{mn}$
$(a^{-2})^2 = a^{-2\times2}=a^{-4}$, $(a^2)^{-4}=a^{2\times(-4)}=a^{-8}$
Multiply terms: $a^{-4} \cdot a^{-8}=a^{-4+(-8)}=a^{-12}$

Step2: Divide by denominator exponent

Use quotient rule: $\frac{x^m}{x^n}=x^{m-n}$
$\frac{a^{-12}}{a^{10}}=a^{-12-10}=a^{-22}$

Step3: Convert to positive exponent

Use negative exponent rule: $x^{-n}=\frac{1}{x^n}$
$a^{-22}=\frac{1}{a^{22}}$

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Step1: Simplify coefficient

Reduce the numerical fraction
$\frac{4}{20}=\frac{1}{5}$

Step2: Simplify $a$-terms

Use quotient rule: $\frac{a^m}{a^n}=a^{m-n}$
$\frac{a^5}{a^4}=a^{5-4}=a^{1}=a$

Step3: Simplify $b$-terms

Use quotient rule and positive exponent conversion
$\frac{b^5}{b^{-2}}=b^{5-(-2)}=b^{7}$

Step4: Combine all simplified terms

Multiply the coefficient and variable terms
$\frac{1}{5} \cdot a \cdot b^7$

Answer:

1. $\frac{1}{a^{22}}$
2. $\frac{ab^7}{5}$