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Chapter 7: Problem 6

Which of the following is equivalent to \(x^{-2}\) ? $$ \begin{array}{r} -x^2 \\ \frac{1}{x^2} \\ \sqrt{x} \end{array} $$ I don't know.

Short Answer

Expert verified
The equivalent of \(x^{-2}\) is \(\frac{1}{x^2}\).

Step by step solution

01

Interpret Negative Exponent

The negative sign in the exponent means that we have to take the reciprocal of the base. So, \(x^{-2}\) can be written as \(1/x^2\). This is due to the rule of exponents that states \(a^{-n} = 1/a^n\). Therefore, \(x^{-2} = 1/x^{2}\).
02

Compare with Options

Next, compare the converted expression with the given options. The first option, \(-x^2\), is not equivalent because the negative sign applies to the whole \(x^2\). Secondly, we have \(\frac{1}{x^2}\), which is the same as our rewritten expression of \(x^{-2}\). The third option, \(\sqrt{x}\), is also not equivalent because it represents the square root of \(x\), not the reciprocal of \(x^2\).

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