Question

select the expression that is equivalent to ((2x^2 + 2)^{-\frac{3}{2}})
answer
(circ) (\frac{1}{sqrt{(2x^2 + 2)^3}}) (circ) (sqrt3{(2x^2 + 2)^2}) (circ) (\frac{1}{sqrt3{(2x^2 + 2)^2}}) (circ) (sqrt{(2x^2 + 2)^3})

Explanation:

Step1: Recall negative exponent rule

A negative exponent means taking the reciprocal, so \(a^{-n}=\frac{1}{a^{n}}\). Here, \(a = 2x^{2}+2\) and \(n=\frac{3}{2}\), so \((2x^{2}+2)^{-\frac{3}{2}}=\frac{1}{(2x^{2}+2)^{\frac{3}{2}}}\).

Step2: Recall fractional exponent rule

A fractional exponent \(\frac{m}{n}\) means the \(n\)-th root of the \(m\)-th power, i.e., \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\). For \((2x^{2}+2)^{\frac{3}{2}}\), \(n = 2\) and \(m = 3\), so \((2x^{2}+2)^{\frac{3}{2}}=\sqrt{(2x^{2}+2)^{3}}\).

Step3: Substitute back

Substituting \((2x^{2}+2)^{\frac{3}{2}}=\sqrt{(2x^{2}+2)^{3}}\) into \(\frac{1}{(2x^{2}+2)^{\frac{3}{2}}}\), we get \(\frac{1}{\sqrt{(2x^{2}+2)^{3}}}\).

Answer:

\(\frac{1}{\sqrt{(2x^{2}+2)^{3}}}\) (the first option)