Question

1. simplify each expression.

a) $(4x^2)(-3x^{-4}y^2)$
b) $dfrac{15xy^4z^{-2}}{20x^{-3}yz^5}$
c) $(3)^{-3}$
d) $dfrac{(2xy^3)^2}{4x^3yz^3}$

Explanation:

Step1: Multiply coefficients, combine $x$ terms

$(4 \times -3)x^{2 + (-4)}y^2 = -12x^{-2}y^2$

Step2: Rewrite negative exponent as reciprocal

$-12x^{-2}y^2 = \frac{-12y^2}{x^2}$

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Step1: Simplify coefficient, combine $x$ terms

$\frac{15}{20}x^{1 - (-3)}y^{4-1}z^{-2-5} = \frac{3}{4}x^{4}y^{3}z^{-7}$

Step2: Rewrite negative exponent as reciprocal

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

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Step1: Rewrite negative exponent as reciprocal

$3^{-3} = \frac{1}{3^3}$

Step2: Evaluate denominator

$\frac{1}{3^3} = \frac{1}{27}$

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Step1: Expand numerator using exponent rule

$\frac{2^2x^2(y^3)^2}{4x^3yz^3} = \frac{4x^2y^6}{4x^3yz^3}$

Step2: Simplify coefficient, combine like terms

$\frac{4}{4}x^{2-3}y^{6-1}z^{-3} = x^{-1}y^{5}z^{-3}$

Step3: Rewrite negative exponents as reciprocals

$x^{-1}y^{5}z^{-3} = \frac{y^5}{xz^3}$

Answer:

a) $\frac{-12y^2}{x^2}$
b) $\frac{3x^4y^3}{4z^7}$
c) $\frac{1}{27}$
d) $\frac{y^5}{xz^3}$