Connect each unsimplified exponential operation to its mathematically equivalent simplifie… — Armed Services Vocational Aptitude Battery (ASVAB)

Connect each unsimplified exponential operation to its mathematically equivalent simplified form.

The correct pairings are: (3a^2)(2a^4) pairs with 6a^6, (12a^6)/(2a^2) pairs with 6a^4, (2a^3)^2 pairs with 4a^6, and 6a^-2 pairs with 6/a^2.

Each pair represents a fundamental rule of exponents applied to a one-step algebraic expression involving both coefficients and variables.

Evaluate the product

(3a^2)(2a^4)

6a^6

By the product rule, multiply coefficients (

3 \times 2 = 6

) and add the exponents of the same base (

2 + 4 = 6

Evaluate the quotient

\frac{12a^6}{2a^2}

6a^4

By the quotient rule, divide coefficients (

12 \div 2 = 6

) and subtract the bottom exponent from the top exponent (

6 - 2 = 4

Evaluate the power

(2a^3)^2

4a^6

By the power of a product rule, apply the exponent to both the coefficient (

2^2 = 4

) and the variable (

3 \times 2 = 6

Evaluate the negative exponent expression

6a^{-2}

\frac{6}{a^2}

By the negative exponent rule,

a^{-2}

becomes

\frac{1}{a^2}

. The coefficient 6 is not raised to the negative power, so it stays in the numerator.

Operations with Variable Exponents

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