Question

6-7. simplify each expression using only positive exponents. homework help a. $(3xy^3)(-2y)$ b. $(x^3y^2)(x^{-2}y^{-2})$ c. $\frac{6x^{-2}}{3x^2}$ d. $(-2x)^{-3}$

Explanation:

Step1: Multiply coefficients and variables

Multiply constants, add exponents of like variables.
$(3xy^3)(-2y) = 3\times(-2) \times x \times y^{3+1}$

Step2: Simplify the expression

Calculate the product and exponent sum.
$= -6xy^4$

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Step1: Multiply like variables, add exponents

Add exponents of $x$ and $y$ separately.
$(x^3y^2)(x^{-2}y^{-2}) = x^{3+(-2)}y^{2+(-2)}$

Step2: Simplify exponents

Simplify the exponent values.
$= x^{1}y^{0} = x$ (since $y^0=1$)

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Step1: Simplify coefficients, subtract exponents

Divide constants, subtract exponents of $x$.
$\frac{6x^{-1}}{3x^2} = \frac{6}{3} \times x^{-1-2}$

Step2: Rewrite with positive exponents

Simplify coefficient and use $x^{-n}=\frac{1}{x^n}$.
$= 2x^{-3} = \frac{2}{x^3}$

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Step1: Apply negative exponent rule

Use $(ab)^{-n}=\frac{1}{(ab)^n}$.
$(-2x)^{-3} = \frac{1}{(-2x)^3}$

Step2: Expand the denominator

Calculate the power of the product.
$= \frac{1}{(-2)^3x^3} = \frac{1}{-8x^3} = -\frac{1}{8x^3}$

Answer:

a. $\boldsymbol{-6xy^4}$
b. $\boldsymbol{x}$
c. $\boldsymbol{\frac{2}{x^3}}$
d. $\boldsymbol{-\frac{1}{8x^3}}$