Exponent Rules

Question

For $x \neq 0$ , which expression equals $\dfrac{(2x^3)^4}{8x^5}$ ?

a. $2x^{7}$
b. $\dfrac{x^{7}}{4}$
c. $2x^{2}$
d. $2x^{17}$

Correct answer

A. $2x^{7}$

Full reasoning

1Apply the power of a product rule: $(2x^3)^4 = 2^4 (x^3)^4$ .
2Apply the power rule to multiply exponents: $2^4 (x^3)^4 = 16x^{12}$ .
3Apply the quotient rule: $\dfrac{16x^{12}}{8x^{5}} = \dfrac{16}{8} x^{12-5} = 2x^{7}$ .

Why each choice is right or wrong

Choice A

Correct. $(2x^3)^4 = 2^4 x^{12} = 16x^{12}$ , and dividing gives $\tfrac{16}{8} x^{12-5} = 2x^{7}$ .

Choice B

This forgets to raise the coefficient $2$ to the fourth power, leaving $2x^{12}$ in the numerator and producing $\tfrac{2}{8}x^{7}$ .

Choice C

This adds the exponents in the power step, turning $(x^3)^4$ into $x^{3+4} = x^{7}$ instead of multiplying to get $x^{12}$ .

Choice D

This adds the exponents when dividing, computing $x^{12+5}$ instead of subtracting to get $x^{12-5}$ .

Related formula

Power of a product and quotient of powers

(ab)^m = a^m b^m, \qquad \frac{a^m}{a^n} = a^{m-n}

Assumptions: The base in the quotient rule is nonzero so the division is defined.

Sources

Authored for this public explainer registry. It has no database question ID and is not copied from a protected or official exam bank.

Elementary Algebra 2e, Section 6.2: Use Multiplication Properties of Exponents — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY 4.0; attribute and avoid verbatim reuse beyond short cited references.

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