Precalculus Examples

${(e^{x})}^{- 3} = \frac{e^{2 x}}{\sqrt{e}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{e}$ as $e^{\frac{1}{2}}$ .

${(e^{x})}^{- 3} = \frac{e^{2 x}}{e^{\frac{1}{2}}}$

Step 2

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

$e^{x \cdot - 3} = \frac{e^{2 x}}{e^{\frac{1}{2}}}$

Step 3

Move $e^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$e^{x \cdot - 3} = e^{2 x} e^{- \frac{1}{2}}$

Step 4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$e^{x \cdot - 3} = e^{2 x - \frac{1}{2}}$

Step 5

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$x \cdot - 3 = 2 x - \frac{1}{2}$

Step 6

Solve for $x$ .

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Step 6.1

Move $- 3$ to the left of $x$ .

$- 3 x = 2 x - \frac{1}{2}$

Step 6.2

Move all terms containing $x$ to the left side of the equation.

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Step 6.2.1

Subtract $2 x$ from both sides of the equation.

$- 3 x - 2 x = - \frac{1}{2}$

Step 6.2.2

Subtract $2 x$ from $- 3 x$ .

$- 5 x = - \frac{1}{2}$

Step 6.3

Divide each term in $- 5 x = - \frac{1}{2}$ by $- 5$ and simplify.

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Step 6.3.1

Divide each term in $- 5 x = - \frac{1}{2}$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{- \frac{1}{2}}{- 5}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $- 5$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{- \frac{1}{2}}{- 5}$

Step 6.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{1}{2}}{- 5}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{1}{2} \cdot \frac{1}{- 5}$

Step 6.3.3.2

Move the negative in front of the fraction.

$x = - \frac{1}{2} (- \frac{1}{5})$

Step 6.3.3.3

Multiply $- \frac{1}{2} (- \frac{1}{5})$ .

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Step 6.3.3.3.1

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{1}{2}) \frac{1}{5}$

Step 6.3.3.3.2

Multiply $\frac{1}{2}$ by $1$ .

$x = \frac{1}{2} \cdot \frac{1}{5}$

Step 6.3.3.3.3

Multiply $\frac{1}{2}$ by $\frac{1}{5}$ .

$x = \frac{1}{2 \cdot 5}$

Step 6.3.3.3.4

Multiply $2$ by $5$ .

$x = \frac{1}{10}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{10}$

Decimal Form:

$x = 0.1$