For all positive values of x, which of the following expressions is equivalent to (8x^6)^{… — SAT

For all positive values of $x$ , which of the following expressions is equivalent to $(8x^6)^{\frac{1}{3}}$ ?

The expression

2x^2

To find an equivalent expression, apply the exponent of

\frac{1}{3}

to each factor inside the parentheses. The expression

(8x^6)^{\frac{1}{3}}

becomes

8^{\frac{1}{3}} \cdot (x^6)^{\frac{1}{3}}

. Since

8^{\frac{1}{3}} = \sqrt[3]{8} = 2

, and

(x^6)^{\frac{1}{3}} = x^{6 \cdot \frac{1}{3}} = x^2

, the simplified expression is

2x^2

Apply the power of a product rule,

(ab)^n = a^n b^n

, to distribute the exponent of

\frac{1}{3}

to both the coefficient and the variable term.

(8x^6)^{\frac{1}{3}} = 8^{\frac{1}{3}} \cdot (x^6)^{\frac{1}{3}}

This separates the numerical coefficient and the variable part to simplify them individually.

Simplify the numerical coefficient by evaluating the cube root of

8

8^{\frac{1}{3}} = 2

The exponent of

\frac{1}{3}

represents the cube root, and

2^3 = 8

Simplify the variable term using the power of a power rule,

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

(x^6)^{\frac{1}{3}} = x^{6 \cdot \frac{1}{3}} = x^2

Multiplying the exponents simplifies the expression.

Multiply the simplified coefficient and variable term together.

2x^2

This gives the final simplified equivalent expression.

Simplifying algebraic expressions with fractional exponents using exponent rules.

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