An engineer is calculating a scaling factor for a fluid dynamics simulation. The calculati… — Armed Services Vocational Aptitude Battery (ASVAB)

An engineer is calculating a scaling factor for a fluid dynamics simulation. The calculation reduces to the following algebraic expression:

\frac{ (3m^3n^{-4})(8m^{-1}n^2) }{ (2m^{-2}n^3)^2 }

Assuming $m$ and $n$ are non-zero variables, what is the fully simplified form of this scaling factor?

The fully simplified expression is

\frac{6m^6}{n^8}

The correct answer is found by simplifying the numerator and denominator independently, then dividing. Multiplying the terms in the numerator gives

24m^2n^{-2}

. Squaring the entire term in the denominator (including the 2) gives

4m^{-4}n^6

. Dividing these expressions yields a coefficient of

6

. Subtracting the exponents gives

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

and

n^{-2 - 6} = n^{-8}

. Finally, converting the negative exponent into a fraction results in

\frac{6m^6}{n^8}

Simplify the numerator using the product rule for exponents.

(3 \times 8) m^{3 + (-1)} n^{-4 + 2} = 24m^2n^{-2}

When multiplying terms with the same base, add their exponents and multiply their numerical coefficients.

Simplify the denominator using the power of a product rule.

2^2(m^{-2 \times 2})(n^{3 \times 2}) = 4m^{-4}n^6

When raising a product to a power, raise the coefficient and multiply every exponent by the outside power.

Divide the simplified numerator by the simplified denominator using the quotient rule.

(\frac{24}{4}) m^{2 - (-4)} n^{-2 - 6} = 6m^6n^{-8}

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

Rewrite the expression with only positive exponents.

\frac{6m^6}{n^8}

A negative exponent means taking the reciprocal of the base, so

n^{-8}

becomes

\frac{1}{n^8}

Operations with Variable Exponents

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