Question

problem #3
$\frac{8x^{-4}y^6}{12x^6}$
problem #4
$(4k^4)^2$
problem #5
$()^3$
problem #6
$2x^2(5x^3)$
problem #7: challenge problem!
$\frac{(x^3y^{-2})^4}{x^2y^3z^0}$
problem #8: challenge problem!
$\frac{(2x^2y^{-2})^2}{2y^3}$

Explanation:

Problem #5

Step1: Apply power of a product rule

$(ab)^n=a^n b^n$
$(2x^2)^3 = 2^3 \cdot (x^2)^3$

Step2: Simplify exponents

$2^3=8$, $(x^2)^3=x^{2 \times 3}=x^6$
$8 \cdot x^6 = 8x^6$

Problem #6

Step1: Multiply coefficients and add exponents

$a^m \cdot a^n=a^{m+n}$
$2x^2(5x^3) = (2 \times 5) \cdot x^{2+3}$

Step2: Simplify terms

$2 \times 5=10$, $x^{2+3}=x^5$
$10 \cdot x^5 = 10x^5$

Problem #7

Step1: Apply power of a product rule

$(a^m)^n=a^{m \times n}$
$\frac{(x^3 y^{-2})^4}{x^2 y^3 z^0} = \frac{x^{3 \times 4} y^{-2 \times 4}}{x^2 y^3 z^0}$

Step2: Simplify exponents

$x^{12} y^{-8}$, $z^0=1$
$\frac{x^{12} y^{-8}}{x^2 y^3}$

Step3: Subtract exponents for like bases

$\frac{a^m}{a^n}=a^{m-n}$
$x^{12-2} y^{-8-3} = x^{10} y^{-11}$

Step4: Rewrite with positive exponents

$a^{-n}=\frac{1}{a^n}$
$x^{10} \cdot \frac{1}{y^{11}} = \frac{x^{10}}{y^{11}}$

Problem #8

Step1: Apply power of a product rule

$(a^m)^n=a^{m \times n}$
$\frac{(2x^2 y^{-2})^2}{2y^3} = \frac{2^2 x^{2 \times 2} y^{-2 \times 2}}{2y^3}$

Step2: Simplify exponents

$2^2=4$, $x^4$, $y^{-4}$
$\frac{4x^4 y^{-4}}{2y^3}$

Step3: Simplify coefficients and subtract exponents

$\frac{4}{2}=2$, $\frac{y^{-4}}{y^3}=y^{-4-3}=y^{-7}$
$2x^4 y^{-7}$

Step4: Rewrite with positive exponents

$2x^4 \cdot \frac{1}{y^7} = \frac{2x^4}{y^7}$

Answer:

Problem #5: $\boldsymbol{8x^6}$
Problem #6: $\boldsymbol{10x^5}$
Problem #7: $\boldsymbol{\frac{x^{10}}{y^{11}}}$
Problem #8: $\boldsymbol{\frac{2x^4}{y^7}}$