Question

which of the following is equivalent to ((4x^2 - 1)^{\frac{-2}{3}})?
a) (sqrt3{(4x^2 - 1)^2})
b) (\frac{1}{sqrt3{(4x^2 - 1)^2}})
c) (\frac{-1}{sqrt3{(4x^2 - 1)^2}})
d) (\frac{1}{sqrt{(4x^2 - 1)^3}})
(lower case letter only) answer

Explanation:

Step1: Recall negative exponent rule

The negative exponent rule states that \(a^{-n}=\frac{1}{a^{n}}\). So, \((4x^{2}-1)^{\frac{-2}{3}}=\frac{1}{(4x^{2}-1)^{\frac{2}{3}}}\).

Step2: Recall fractional exponent rule

The fractional exponent rule \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\). So, \((4x^{2}-1)^{\frac{2}{3}}=\sqrt[3]{(4x^{2}-1)^{2}}\).

Step3: Substitute back

Substituting \((4x^{2}-1)^{\frac{2}{3}}=\sqrt[3]{(4x^{2}-1)^{2}}\) into \(\frac{1}{(4x^{2}-1)^{\frac{2}{3}}}\), we get \(\frac{1}{\sqrt[3]{(4x^{2}-1)^{2}}}\), which matches option b.

Answer:

b