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Question : 26 of 86

Marks: +1, -0

\left[1-\left\{1-(1-x^{2})^{-1}\right\}^{-1}\right]^{-\frac{1}{2}}

simplified the term

0
$x$
$x^{-1}$
1

Solution:

Concept:

Simplify the nested expression step by step using algebraic manipulation and exponent rules.

Explanation:

Start with:

\left[1-\left\{1-(1-x^{2})^{-1}\right\}^{-1}\right]^{-\frac{1}{2}}

Simplify inner bracket:

1-(1-x^{2})^{-1} = 1 - \frac{1}{1-x^{2}} = \frac{1-x^{2}-1}{1-x^{2}} = \frac{-x^{2}}{1-x^{2}}

Then invert:

\left\{ \frac{-x^{2}}{1-x^{2}} \right\}^{-1} = -\frac{1-x^{2}}{x^{2}}

Now compute:

1 - \left(-\frac{1-x^{2}}{x^{2}}\right) = 1 + \frac{1-x^{2}}{x^{2}} = \frac{x^{2}+1-x^{2}}{x^{2}} = \frac{1}{x^{2}}

Finally raise to

-\frac{1}{2}

:

\left(\frac{1}{x^{2}}\right)^{-\frac{1}{2}} = (x^{-2})^{-\frac{1}{2}} = x^{1} = x

Valid when

x \neq 0

and

1-x^{2} \neq 0

.

Answer:

x

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