A student is asked to simplify the rational algebraic expression shown below. \frac{(3a^2b… — Armed Services Vocational Aptitude Battery (ASVAB)

A student is asked to simplify the rational algebraic expression shown below.

\frac{(3a^2b^{-3})^2}{3a^{-1}b^{-4}}

What is the fully simplified form of this expression with positive exponents?

\frac{3a^5}{b^2}

To simplify the expression, first simplify the numerator by squaring every factor inside the parentheses:

(3)^2 \cdot (a^2)^2 \cdot (b^{-3})^2

, which yields

9a^4b^{-6}

. Next, divide this result by the denominator

3a^{-1}b^{-4}

. Divide the coefficients:

9 \div 3 = 3

. Then, apply the quotient rule by subtracting the exponents of like bases: for

a

4 - (-1) = 5

; for

b

-6 - (-4) = -2

. This gives

3a^5b^{-2}

. Finally, rewrite the expression with positive exponents by moving

b^{-2}

to the denominator, resulting in

\frac{3a^5}{b^2}

Apply the power of a product and power of a power rules to the numerator:

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

9a^4b^{-6}

The exponent outside the parentheses applies to the coefficient and is multiplied by each variable's exponent.

Substitute the expanded numerator back into the fraction.

\frac{9a^4b^{-6}}{3a^{-1}b^{-4}}

This sets up the expression for the quotient rule.

Divide the numerical coefficients.

9 \div 3 = 3

Coefficients are divided normally.

Apply the quotient rule by subtracting the denominator's exponents from the numerator's exponents for bases

a

and

b

a^{4 - (-1)} = a^5

and

b^{-6 - (-4)} = b^{-2}

When dividing like bases, exponents are subtracted. Care must be taken with double negative signs.

Combine the results and convert any negative exponents to positive exponents by moving them to the denominator.

3a^5b^{-2} = \frac{3a^5}{b^2}

Standard mathematical notation requires final answers to be expressed using positive exponents.

Applying exponent rules (power, product, and quotient) with numerical coefficients and negative integers.

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