Which of the following is equivalent to the expression \frac{(x^2)^3}{x^{-2}} for all posi… — SAT

Which of the following is equivalent to the expression $\frac{(x^2)^3}{x^{-2}}$ for all positive values of $x$ ?

A
$x^3$
B
$x^4$
$x^8$ Answer
D
$x^7$

Answer

The expression

x^8

is equivalent to the given expression.

The correct expression is

x^8

. First, simplify the numerator using the power of a power rule:

(x^2)^3 = x^{2 \cdot 3} = x^6

. Next, divide by the denominator using the quotient rule of exponents, which requires subtracting the exponent in the denominator from the exponent in the numerator:

x^6 / x^{-2} = x^{6 - (-2)} = x^{6 + 2} = x^8

Step-by-Step Solution

Apply the power of a power rule to the numerator:

(x^a)^b = x^{a \cdot b}

(x^2)^3 = x^{2 \cdot 3} = x^6

To raise an exponential term to a power, multiply the exponents.

Apply the quotient rule for exponents:

\frac{x^a}{x^b} = x^{a - b}

\frac{x^6}{x^{-2}} = x^{6 - (-2)} = x^{6 + 2} = x^8

To divide powers with the same base, subtract the exponent in the denominator from the exponent in the numerator.

Key Concept

Simplifying algebraic expressions containing negative and positive exponents using the rules of exponents.

Question

Answer

Step-by-Step Solution

Key Concept

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