Calculus Examples

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

Step 1

Cancel the common factor of $e^{x}$ and $e^{x - 2}$ .

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Step 1.1

Factor $e^{x - 2}$ out of $e^{x}$ .

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

Step 1.2

Cancel the common factors.

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Step 1.2.1

Multiply by $1$ .

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

Step 1.2.2

Cancel the common factor.

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

Step 1.2.3

Rewrite the expression.

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

Step 1.2.4

Divide $e^{x - (x - 2)}$ by $1$ .

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

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Multiply $- 1$ by $- 2$ .

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

Step 3

Simplify by adding terms.

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Step 3.1

Subtract $x$ from $x$ .

${(e^{0 + 2})}^{- \frac{1}{2}}$

Step 3.2

Simplify the expression.

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Step 3.2.1

Add $0$ and $2$ .

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

Step 3.2.2

Multiply the exponents in ${(e^{2})}^{- \frac{1}{2}}$ .

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Step 3.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.2.2.2

Cancel the common factor of $2$ .

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Step 3.2.2.2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 3.2.2.2.2

Cancel the common factor.

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

Step 3.2.2.2.3

Rewrite the expression.

$e^{- 1}$

Step 4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{e}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{e}$

Decimal Form:

$0.36787944 \dots$