Which of the following expressions is equivalent to (8x^6)^{\frac{2}{3}} \cdot (4x^{-2})^{… — SAT

Which of the following expressions is equivalent to $(8x^6)^{\frac{2}{3}} \cdot (4x^{-2})^{-\frac{1}{2}}$ for all positive values of $x$ ?

2x^5

To find the equivalent expression, simplify each part of the product. The first part,

(8x^6)^{\frac{2}{3}}

, simplifies to

8^{\frac{2}{3}} \cdot x^{6 \cdot \frac{2}{3}} = 4x^4

. The second part,

(4x^{-2})^{-\frac{1}{2}}

, simplifies to

4^{-\frac{1}{2}} \cdot x^{-2 \cdot -\frac{1}{2}} = \frac{1}{2}x

. Multiplying these two simplified expressions gives

4x^4 \cdot \frac{1}{2}x = 2x^5

. This matches the expression

2x^5

Simplify the first term,

(8x^6)^{\frac{2}{3}}

, by distributing the exponent to both the coefficient and the variable.

(8x^6)^{\frac{2}{3}} = 8^{\frac{2}{3}} \cdot (x^6)^{\frac{2}{3}} = (\sqrt[3]{8})^2 \cdot x^{6 \cdot \frac{2}{3}} = 2^2 \cdot x^4 = 4x^4

The power of a product rule states that

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

, and the power of a power rule states that

(a^m)^n = a^{mn}

Simplify the second term,

(4x^{-2})^{-\frac{1}{2}}

, by distributing the exponent to both the coefficient and the variable.

(4x^{-2})^{-\frac{1}{2}} = 4^{-\frac{1}{2}} \cdot (x^{-2})^{-\frac{1}{2}} = \frac{1}{\sqrt{4}} \cdot x^{-2 \cdot -\frac{1}{2}} = \frac{1}{2}x^1 = \frac{1}{2}x

Applying the same power rules, negative exponent properties, and multiplying negative exponents yields a positive exponent:

-2 \cdot -\frac{1}{2} = 1

Multiply the simplified expressions obtained in Step 1 and Step 2.

4x^4 \cdot \frac{1}{2}x = (4 \cdot \frac{1}{2}) \cdot (x^4 \cdot x^1) = 2x^{4+1} = 2x^5

Multiply the coefficients and add the exponents of the same base variable according to the product rule

a^m \cdot a^n = a^{m+n}

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

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