Which of the following is equivalent to the expression (3m^{-2}n^4)^2(2m^5n^0)? — Armed Services Vocational Aptitude Battery (ASVAB)

Which of the following is equivalent to the expression $(3m^{-2}n^4)^2(2m^5n^0)$ ?

The simplified expression is

18mn^8

The correct answer is derived by first squaring each part of the initial term, making it

9m^{-4}n^8

. The second term simplifies to

2m^5

because

n^0 = 1

. Multiplying these together requires multiplying the coefficients (

9 \times 2 = 18

) and adding the exponents for

m

(

-4 + 5 = 1

), resulting in

18mn^8

Simplify the first factor by applying the power of 2 to the coefficient and the variable exponents.

(3)^2(m^{-2})^2(n^4)^2 = 9m^{-4}n^8

When a product is raised to a power, the exponent must be applied to every factor inside the parentheses, multiplying the variable exponents.

Simplify the second factor using the zero exponent rule.

2m^5(1) = 2m^5

Any non-zero term raised to the power of 0 equals 1, so

n^0 = 1

Multiply the two simplified factors together.

(9m^{-4}n^8)(2m^5) = (9 \cdot 2)(m^{-4} \cdot m^5)(n^8)

To multiply terms, multiply the numerical coefficients and add the exponents of like bases.

Perform the final arithmetic for the coefficients and exponents.

18m^{-4+5}n^8 = 18m^1n^8 = 18mn^8

Evaluating

9 \cdot 2 = 18

and combining the exponents of

m

gives the final simplified expression.

Operations with Variable Exponents

Question