Simplify the following algebraic expression, assuming all variables are non-zero: \frac{(3… — Armed Services Vocational Aptitude Battery (ASVAB)

Simplify the following algebraic expression, assuming all variables are non-zero:

\frac{(3m^2n^4)^2}{3m^5n^3}

Which of the following represents the fully simplified form?

The fully simplified expression is

\frac{3n^5}{m}

The correct expression is derived by properly applying the power rules. First, squaring the numerator

(3m^2n^4)^2

yields

9m^4n^8

. Then, dividing this by the denominator

3m^5n^3

requires dividing the coefficients (

9/3 = 3

) and subtracting the exponents of corresponding variables (

4-5 = -1

for

m

, and

8-3 = 5

for

n

). Finally, rewriting

3m^{-1}n^5

with only positive exponents gives

\frac{3n^5}{m}

Simplify the numerator using the power of a product rule.

(3m^2n^4)^2 = 3^2 \cdot (m^2)^2 \cdot (n^4)^2 = 9m^4n^8

When a product is raised to a power, each factor inside the parentheses must be raised to that power.

Substitute the simplified numerator back into the fraction.

\frac{9m^4n^8}{3m^5n^3}

This sets up the expression for applying the quotient rules.

Divide the numerical coefficients.

\frac{9}{3} = 3

Constants are simplified through standard division.

Apply the quotient rule for exponents to the variables

m

and

n

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

and

n^{8-3} = n^5

When dividing terms with the same base, subtract the exponent in the denominator from the exponent in the numerator.

Combine the results and rewrite without negative exponents.

3m^{-1}n^5 = \frac{3n^5}{m}

A negative exponent indicates the reciprocal of the base, moving the term to the denominator.

Exponent rules including power of a power, power of a product, and quotient of powers.

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