Question

Explaining a First Move
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

Answer 1

Both are correct — the two approaches are equivalent because (a/b)^3 = a^3 / b^3 and because of the laws of exponents.

Show both ways (assuming m and n ≠ 0):

Tyrone’s way (raise numerator and denominator to the 3rd power):
((m^2 n^−3)/(m^6 n^−1))^3 = (m^2 n^−3)^3 / (m^6 n^−1)^3
= m^(2·3) n^(−3·3) / m^(6·3) n^(−1·3)
= m^6 n^−9 / m^18 n^−3
= m^(6−18) n^(−9−(−3)) = m^−12 n^−6 = 1/(m^12 n^6).

Alisha’s way (simplify inside first, using quotient rules):
(m^2 n^−3)/(m^6 n^−1) = m^(2−6) n^(−3−(−1)) = m^−4 n^−2,
then ((m^−4 n^−2))^3 = m^(−4·3) n^(−2·3) = m^−12 n^−6 = 1/(m^12 n^6).

Either method yields the same result; simplifying inside first can be slightly quicker, but both are valid.