Question

Using the properties of exponents, which expression is equivalent to x^(-2) * x^(1/2)?

The answer options are:

sqrt(x)

1/x

1/sqrt(x)

x^(1/2)

Answer 1

To simplify the expression \( x^{-2} * x^{1/2} \), we can use the property of exponents that states \( a^m * a^n = a^{m+n} \).

Applying this property here:

\[
x^{-2} * x^{1/2} = x^{-2 + 1/2}
\]

Now, let's simplify the exponent:

\[
-2 + \frac{1}{2} = -\frac{4}{2} + \frac{1}{2} = -\frac{4 - 1}{2} = -\frac{3}{2}
\]

So, we have:

\[
x^{-2} * x^{1/2} = x^{-\frac{3}{2}}
\]

Using the property \( a^{-m} = \frac{1}{a^m} \), we can rewrite \( x^{-\frac{3}{2}} \) as:

\[
x^{-\frac{3}{2}} = \frac{1}{x^{\frac{3}{2}}}
\]

Now, \( x^{\frac{3}{2}} \) can be rewritten in radical form:

\[
x^{\frac{3}{2}} = (x^{\frac{1}{2}})^3 = (\sqrt{x})^3 = x\sqrt{x}
\]

Thus, we have:

\[
x^{-\frac{3}{2}} = \frac{1}{x\sqrt{x}} = \frac{1}{\sqrt{x^3}}
\]

However, for our answer options, we can express it simply equivalently as:

\[
\frac{1}{\sqrt{x^3}} \quad \text{or} \quad \frac{1}{(\sqrt{x})^3}
\]

Among your answer options:

- \( \sqrt{x} \)
- \( \frac{1}{x} \)
- \( \frac{1}{\sqrt{x}} \)
- \( x^{1/2} \)

None of these match exactly with \( x^{-\frac{3}{2}} \), but since the options are limited, the closest one would be:

\[
\frac{1}{\sqrt{x}}.
\]

**Final answer:** \( \frac{1}{\sqrt{x}} \).