Trigonometry Examples

$x = \sqrt[10]{e^{- 5}}$

Step 1

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

$x = \sqrt[10]{\frac{1}{e^{5}}}$

Step 2

Rewrite $1$ as $1^{5}$ .

$x = \sqrt[10]{\frac{1^{5}}{e^{5}}}$

Step 3

Rewrite $\frac{1^{5}}{e^{5}}$ as ${(\frac{1}{e})}^{5}$ .

$x = \sqrt[10]{{(\frac{1}{e})}^{5}}$

Step 4

Rewrite $\sqrt[10]{{(\frac{1}{e})}^{5}}$ as $\sqrt{\sqrt[5]{{(\frac{1}{e})}^{5}}}$ .

$x = \sqrt{\sqrt[5]{{(\frac{1}{e})}^{5}}}$

Step 5

Pull terms out from under the radical, assuming real numbers.

$x = \sqrt{\frac{1}{e}}$

Step 6

Rewrite $\sqrt{\frac{1}{e}}$ as $\frac{\sqrt{1}}{\sqrt{e}}$ .

$x = \frac{\sqrt{1}}{\sqrt{e}}$

Step 7

Any root of $1$ is $1$ .

$x = \frac{1}{\sqrt{e}}$

Step 8

Multiply $\frac{1}{\sqrt{e}}$ by $\frac{\sqrt{e}}{\sqrt{e}}$ .

$x = \frac{1}{\sqrt{e}} \cdot \frac{\sqrt{e}}{\sqrt{e}}$

Step 9

Combine and simplify the denominator.

Tap for more steps...

Step 9.1

Multiply $\frac{1}{\sqrt{e}}$ by $\frac{\sqrt{e}}{\sqrt{e}}$ .

$x = \frac{\sqrt{e}}{\sqrt{e} \sqrt{e}}$

Step 9.2

Raise $\sqrt{e}$ to the power of $1$ .

$x = \frac{\sqrt{e}}{{\sqrt{e}}^{1} \sqrt{e}}$

Step 9.3

Raise $\sqrt{e}$ to the power of $1$ .

$x = \frac{\sqrt{e}}{{\sqrt{e}}^{1} {\sqrt{e}}^{1}}$

Step 9.4

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

$x = \frac{\sqrt{e}}{{\sqrt{e}}^{1 + 1}}$

Step 9.5

Add $1$ and $1$ .

$x = \frac{\sqrt{e}}{{\sqrt{e}}^{2}}$

Step 9.6

Rewrite ${\sqrt{e}}^{2}$ as $e$ .

Tap for more steps...

Step 9.6.1

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

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

Step 9.6.2

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

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

Step 9.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \frac{\sqrt{e}}{e^{\frac{2}{2}}}$

Step 9.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.6.4.1

Cancel the common factor.

$x = \frac{\sqrt{e}}{e^{\frac{2}{2}}}$

Step 9.6.4.2

Rewrite the expression.

$x = \frac{\sqrt{e}}{e^{1}}$

Step 9.6.5

Simplify.

$x = \frac{\sqrt{e}}{e}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{e}}{e}$

Decimal Form:

$x = 0.60653065 \dots$