Multiple Choice
Simplify each expression, but don’t evaluate.
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6. Exponents, Polynomials, and Polynomial Functions
Intro to the Power Rules
Multiple Choice
Simplify each expression.
(6m^2 n^3 )^2/(-4m^7 n^5 )^3
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Verified step by step guidance1
Start by applying the power of a product rule to both the numerator and the denominator. This means raising each factor inside the parentheses to the given exponent: write \((6m^2n^3)^2\) as \(6^2 \cdot (m^2)^2 \cdot (n^3)^2\) and \((-4m^7n^5)^3\) as \((-4)^3 \cdot (m^7)^3 \cdot (n^5)^3\).
Simplify each part by calculating the powers: \$6^2\(, \)(-4)^3$, and use the power of a power rule for the variables, which states \((a^m)^n = a^{m \cdot n}\). So, \((m^2)^2 = m^{2 \cdot 2}\), \((n^3)^2 = n^{3 \cdot 2}\), \((m^7)^3 = m^{7 \cdot 3}\), and \((n^5)^3 = n^{5 \cdot 3}\).
Rewrite the expression with these simplified powers: the numerator becomes \$6^2 m^{4} n^{6}\( and the denominator becomes \)(-4)^3 m^{21} n^{15}$.
Next, write the entire expression as a fraction: \(\frac{6^2 m^{4} n^{6}}{(-4)^3 m^{21} n^{15}}\). Then, separate the constants and the variables: \(\frac{6^2}{(-4)^3} \cdot \frac{m^{4}}{m^{21}} \cdot \frac{n^{6}}{n^{15}}\).
Finally, apply the quotient rule for exponents, which states \(\frac{a^m}{a^n} = a^{m-n}\), to simplify the variables: \(m^{4-21} = m^{-17}\) and \(n^{6-15} = n^{-9}\). Combine the constants and variables to write the simplified expression.