An algebraic expression is given as: \frac{(3a^4b^{-2})^2 (2a^{-3}b^5)}{6a^2b} What is the… — Armed Services Vocational Aptitude Battery (ASVAB)

An algebraic expression is given as:

\frac{(3a^4b^{-2})^2 (2a^{-3}b^5)}{6a^2b}

What is the fully simplified form of this expression for all non-zero values of $a$ and $b$ ?

The simplified form of the expression is

3a^3

The expression is correctly simplified by first applying the power to all terms inside the parentheses (resulting in

9a^8b^{-4}

), then multiplying by the adjacent term in the numerator (resulting in

18a^5b

), and finally dividing by the denominator (giving

3a^3

Simplify the squared term in the numerator using the power of a product and power of a power rules.

9a^8b^{-4}

When a product is raised to a power, raise each factor to that power (

3^2 = 9

). For variables, multiply the inner and outer exponents:

(a^4)^2 = a^8

and

(b^{-2})^2 = b^{-4}

Multiply the simplified term by the second factor in the numerator,

(2a^{-3}b^5)

18a^5b

Multiply the coefficients (

9 \times 2 = 18

). For variables with the same base, add their exponents:

a^{8+(-3)} = a^5

and

b^{-4+5} = b^1 = b

Divide the resulting numerator by the denominator,

6a^2b

3a^3

Divide the coefficients (

18 \div 6 = 3

). For variables with the same base, subtract the denominator's exponent from the numerator's exponent:

a^{5-2} = a^3

. The terms

b^1

and

b^1

cancel out since

b^{1-1} = b^0 = 1

Operations with Variable Exponents

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