Which of the following expressions is equivalent to (2x^3)^4 for all values of x? — SAT

Which of the following expressions is equivalent to $(2x^3)^4$ for all values of $x$ ?

16x^{12}

The correct answer is the expression

16x^{12}

. When simplifying

(2x^3)^4

, we apply the exponent

4

to both the coefficient

2

and the term

x^3

, resulting in

2^4(x^3)^4

. Evaluating

2^4

gives

16

. Applying the power of a power rule to

(x^3)^4

requires multiplying the exponents (

3 \times 4

), which gives

x^{12}

. Combining these yields

16x^{12}

Apply the power of a product property to distribute the outer exponent of

4

to both factors inside the parentheses: the coefficient

2

and the variable term

x^3

(2x^3)^4 = 2^4 \times (x^3)^4

The power of a product rule states that

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

Evaluate the constant coefficient

2^4

2^4 = 2 \times 2 \times 2 \times 2 = 16

Evaluating the base raised to the exponent.

Apply the power of a power property to simplify the variable term

(x^3)^4

(x^3)^4 = x^{3 \times 4} = x^{12}

The power of a power rule states that

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

Combine the simplified coefficient and variable parts to find the final equivalent expression.

16x^{12}

Combining the results of the coefficient simplification and the variable simplification yields the final term.

Exponent rules, specifically the power of a product property

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

and the power of a power property

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

Estimated Time:45s

Question