Expand using index laws 4 and 5

Powers of products and quotients

An outside power applies to every factor inside a product or quotient. Multiply each existing index by the outside power and calculate the numerical coefficient separately. Here n is a positive integer. Denominators must be nonzero

1. Raise every product factor The outside power applies to the coefficient and every factor inside the brackets \(\displaystyle \begin{gathered}(2xy^2)^3=2^3x^3y^6\\=8x^3y^6\end{gathered}\)
2. Raise numerator and denominator For a quotient, apply the outside power separately to every factor on top and bottom. Every denominator must be nonzero; here b ≠ 0. \(\displaystyle \begin{gathered}(\frac{3a^2}{b})^2=\frac{3^2a^4}{b^2}\\=\frac{9a^4}{b^2}\end{gathered}\)
3. Multiply existing indices An index already on a factor is multiplied by the outside power \(\displaystyle \left(x^{3}\right)^{2}=x^{3\times2}=x^{6}\)
4. Track a negative factor The sign of the numerical coefficient follows the parity of the outside power. This does not fix the sign of an expression when its variables are unknown \(\displaystyle \left(-2x\right)^{3}=(-2)^{3}x^{3}=-8x^{3}\)

Worked example

Expand the quotient. Assume c ≠ 0.

1. \(\displaystyle \begin{gathered}(\frac{2a^3b}{5c^2})^2\\=\frac{2^2(a^3)^2b^2}{5^2(c^2)^2}\end{gathered}\)Apply the outside power to every factor in the numerator and denominator
2. \(\displaystyle 2^{2}=4,\quad5^{2}=25\)Evaluate the coefficient powers in the numerator and denominator
3. \(\displaystyle \begin{gathered}3\times2=6\\1\times2=2\\2\times2=4\end{gathered}\)Multiply each existing index by the outside power
4. \(\displaystyle \left(\frac{2a^{3}b}{5c^{2}}\right)^{2}=\frac{4a^{6}b^{2}}{25c^{4}}\)Combine the evaluated coefficients and the new indices

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