Simplify the following algebraic expression: \left( \frac{12p^4 q^{-3}}{3p^{-2} q^2} \righ… — Armed Services Vocational Aptitude Battery (ASVAB)

Simplify the following algebraic expression:

\left( \frac{12p^4 q^{-3}}{3p^{-2} q^2} \right)^{-1}

What is the fully simplified form of this expression?

The expression simplifies to

\frac{q^5}{4p^6}

The correct answer is derived by first simplifying the inner fraction using the quotient rule for exponents, resulting in

4p^6q^{-5}

. Applying the outer negative exponent of

-1

takes the reciprocal of each factor, leading to

4^{-1}p^{-6}q^5

, which is written with positive exponents as

\frac{q^5}{4p^6}

Divide the numerical coefficients.

12 \div 3 = 4

Coefficients follow standard division rules.

Simplify the variable

p

using the quotient rule for exponents.

p^4 \div p^{-2} = p^{4 - (-2)} = p^6

When dividing like bases, subtract the exponent in the denominator from the exponent in the numerator.

Simplify the variable

q

using the quotient rule for exponents.

q^{-3} \div q^2 = q^{-3 - 2} = q^{-5}

Subtracting a positive exponent is mathematically equivalent to adding its negative value.

Combine the simplified parts into a single expression inside the parentheses.

4p^6q^{-5}

This represents the fully simplified value of the fraction before applying the outer exponent.

Apply the outer exponent of

-1

to every factor in the expression.

(4p^6q^{-5})^{-1} = 4^{-1} \cdot (p^6)^{-1} \cdot (q^{-5})^{-1}

The power of a product rule states that

(abc)^x = a^x b^x c^x

Simplify the resulting exponents and write with positive exponents.

4^{-1}p^{-6}q^5 = \frac{q^5}{4p^6}

A negative exponent indicates the reciprocal of the base (

x^{-n} = 1/x^n

Simplifying algebraic fractions using the quotient, product, and negative power rules for exponents.

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