Question

question 9
use the properties of exponents to simplify each expression. write all answers with positive exponents only. (assume variables are nonzero.)
\\(\left(\frac{6ax}{2a^4x^{-2}}\
ight)^3\\)

question 10
use the properties of exponents to simplify each expression. write all answers with positive exponents only. (assume variables are nonzero.)
\\(\left(\frac{x^{-7}y^3z^5}{x^{-2}y^7z^4}\
ight)^0\\)

Explanation:

Step1: Simplify the fraction inside the parentheses

First, simplify the coefficients and like variables using exponent rules $\frac{x^m}{x^n}=x^{m-n}$ and $\frac{a}{b}=\frac{a\div c}{b\div c}$ for coefficients:
$\frac{6ax}{2a^4x^{-2}} = \frac{6}{2} \cdot \frac{a}{a^4} \cdot \frac{x}{x^{-2}} = 3 \cdot a^{1-4} \cdot x^{1-(-2)} = 3a^{-3}x^{3}$

Step2: Apply the power of a power rule

Raise the simplified expression to the 3rd power using $(x^m)^n=x^{mn}$ and $(ab)^n=a^nb^n$:
$(3a^{-3}x^{3})^3 = 3^3 \cdot (a^{-3})^3 \cdot (x^{3})^3 = 27a^{-9}x^{9}$

Step3: Rewrite with positive exponents

Use $x^{-n}=\frac{1}{x^n}$ to convert negative exponents:
$27a^{-9}x^{9} = \frac{27x^{9}}{a^{9}}$

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Step4: Simplify the fraction inside the parentheses

Simplify the expression inside using $\frac{x^m}{x^n}=x^{m-n}$:
$\frac{x^{-7}y^3z^5}{x^{-2}y^7z^4} = x^{-7-(-2)}y^{3-7}z^{5-4} = x^{-5}y^{-4}z^{1}$

Step5: Apply the zero exponent rule

Use the rule $x^0=1$ for any non-zero $x$:
$(x^{-5}y^{-4}z)^{0} = 1$

Answer:

Question 9: $\frac{27x^{9}}{a^{9}}$
Question 10: $1$