Precalculus Examples

e^{(- \frac{1}{2}) \ln (x)}

$e^{(- \frac{1}{2}) \ln (x)}$

Step 1

Multiply

(- \frac{1}{2}) \ln (x)

$(- \frac{1}{2}) \ln (x)$ .

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Step 1.1

Reorder

\ln (x)

$\ln (x)$ and

\frac{1}{2}

$\frac{1}{2}$ .

e^{- (\frac{1}{2} \ln (x))}

$e^{- (\frac{1}{2} \ln (x))}$

Step 1.2

Simplify

\frac{1}{2} \ln (x)

$\frac{1}{2} \ln (x)$ by moving

\frac{1}{2}

$\frac{1}{2}$ inside the logarithm.

e^{- \ln (x^{\frac{1}{2}})}

$e^{- \ln (x^{\frac{1}{2}})}$

e^{- \ln (x^{\frac{1}{2}})}

$e^{- \ln (x^{\frac{1}{2}})}$

Step 2

Simplify

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

$- \ln (x^{\frac{1}{2}})$ by moving

- 1

$- 1$ inside the logarithm.

e^{\ln ({(x^{\frac{1}{2}})}^{- 1})}

$e^{\ln ({(x^{\frac{1}{2}})}^{- 1})}$

Step 3

Exponentiation and log are inverse functions.

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

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

Step 4

Multiply the exponents in

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

${(x^{\frac{1}{2}})}^{- 1}$ .

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Step 4.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

x^{\frac{1}{2} \cdot - 1}

$x^{\frac{1}{2} \cdot - 1}$

Step 4.2

Combine

\frac{1}{2}

$\frac{1}{2}$ and

- 1

$- 1$ .

x^{\frac{- 1}{2}}

$x^{\frac{- 1}{2}}$

Step 4.3

Move the negative in front of the fraction.

x^{- \frac{1}{2}}

$x^{- \frac{1}{2}}$

x^{- \frac{1}{2}}

$x^{- \frac{1}{2}}$

Step 5

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

\frac{1}{x^{\frac{1}{2}}}

$\frac{1}{x^{\frac{1}{2}}}$

e^{(- \frac{1}{2}) \ln (x)}

$e^{(- \frac{1}{2}) \ln (x)}$

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[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$