Which of the following is equivalent to the expression \frac{(5^3)^4}{5^2}? — ACT

Which of the following is equivalent to the expression $\frac{(5^3)^4}{5^2}$ ?

The correct simplified expression is

5^{10}

The expression can be simplified using basic exponent laws. First, using the power of a power rule, the numerator

(5^3)^4

simplifies to

5^{3 \times 4} = 5^{12}

. Then, using the quotient rule, dividing

5^{12}

5^2

simplifies to

5^{12 - 2} = 5^{10}

. This corresponds to the option containing

5^{10}

Simplify the numerator using the power of a power rule:

(a^m)^n = a^{m \times n}

(5^3)^4 = 5^{3 \times 4} = 5^{12}

When raising a power to another power, multiply the exponents.

Simplify the fraction using the quotient rule:

\frac{a^m}{a^n} = a^{m - n}

\frac{5^{12}}{5^2} = 5^{12 - 2} = 5^{10}

When dividing exponential expressions with the same base, subtract the exponent of the denominator from the exponent of the numerator.

Applying exponent rules (power of a power rule and quotient rule) to simplify exponential expressions.

Alternatively, you can write out the terms as repeated multiplication:

(5^3)^4 = 5^3 \times 5^3 \times 5^3 \times 5^3 = 5^{12}

, and then divide by

5^2

by cancelling out two factors of 5, which leaves ten factors of 5, or

5^{10}

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