Question

Tyrone claims that the first step to simplify this expression is to raise the numerator and denominator to the third power. Alisha claims that the first step to simplify is to apply the quotient of powers. Who is correct? Explain.
(StartFraction m squared n Superscript negative 3 Baseline Over m Superscript 6 Baseline n Superscript negative 1 Baseline EndFraction) cubed
wrote one paragraph

Answer 1

Alisha is correct in her approach to simplifying the expression \(\left(\frac{m^2 n^{-3}}{m^6 n^{-1}}\right)^3\). The first step should be to apply the quotient of powers property, which states that when dividing like bases, we subtract the exponents. This allows us to simplify the fraction before raising it to the third power. By using the quotient of powers, we combine the terms in the numerator and denominator: for \(m\), we have \(m^{2-6} = m^{-4}\), and for \(n\), we have \(n^{-3 - (-1)} = n^{-2}\). Therefore, the expression simplifies to \(\left(\frac{m^{-4}}{n^2}\right)^3\) before we apply the exponent of 3, further simplifying the terms effectively. Tyrone's suggestion to immediately raise the entire fraction to the third power can lead to unnecessary complexity and potential errors in handling the powers.