Pair each algebraic operation on the left with its correctly simplified expression on the … — Armed Services Vocational Aptitude Battery (ASVAB)

Pair each algebraic operation on the left with its correctly simplified expression on the right. Assume all variables represent non-zero real numbers.

$(3m^4 n^2)(4m^2 n^5)$ $12m^6 n^7$
$\frac{12m^8 n^3}{4m^2 n^{-4}}$ $3m^6 n^7$
$(2m^3 n^2)^3$ $8m^9 n^6$
$\frac{24m^4 n^2}{3m^5 n^{-1}}$ $\frac{8n^3}{m}$

Answer

The correct matches require applying the product, quotient, and power rules of exponents systematically to each expression.

Each expression is correctly simplified by identifying the primary exponent rule required (product, quotient, or power) and applying it to both the numerical coefficients and the variables' exponents.

Step-by-Step Solution

Evaluate

(3m^4 n^2)(4m^2 n^5)

using the Product Rule.

12m^6 n^7

Multiply coefficients (3 * 4 = 12) and add exponents of corresponding variables (

m^{4+2}

n^{2+5}

Evaluate

\frac{12m^8 n^3}{4m^2 n^{-4}}

using the Quotient Rule.

3m^6 n^7

Divide coefficients (12 / 4 = 3) and subtract denominator exponents from numerator exponents (

m^{8-2}

n^{3-(-4)}

Evaluate

(2m^3 n^2)^3

using the Power of a Product Rule.

8m^9 n^6

Cube the coefficient (

2^3 = 8

) and multiply inner exponents by the outer exponent (

m^{3 \times 3}

n^{2 \times 3}

Evaluate

\frac{24m^4 n^2}{3m^5 n^{-1}}

using the Quotient Rule and Negative Exponent Rule.

\frac{8n^3}{m}

Divide coefficients (24 / 3 = 8), subtract exponents (

m^{4-5} = m^{-1}

n^{2-(-1)} = n^3

), and rewrite the negative exponent in the denominator.

Key Concept

Applying fundamental exponent laws (product rule, quotient rule, power of a power) and properly handling numerical coefficients separately from variable exponents.

Estimated Time:1m 30s

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