Calculus Examples

$lim_{x \to 8} \sqrt{\frac{{(- 3)}^{x - 1}}{\sqrt{x}}}$

Step 1

Move the limit under the radical sign.

$\sqrt{lim_{x \to 8} \frac{{(- 3)}^{x - 1}}{\sqrt{x}}}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

$\sqrt{\frac{lim_{x \to 8} {(- 3)}^{x - 1}}{lim_{x \to 8} \sqrt{x}}}$

Step 3

Move the limit into the exponent.

$\sqrt{\frac{{(- 3)}^{lim_{x \to 8} x - 1}}{lim_{x \to 8} \sqrt{x}}}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\sqrt{\frac{{(- 3)}^{lim_{x \to 8} x - lim_{x \to 8} 1}}{lim_{x \to 8} \sqrt{x}}}$

Step 5

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$\sqrt{\frac{{(- 3)}^{lim_{x \to 8} x - 1 \cdot 1}}{lim_{x \to 8} \sqrt{x}}}$

Step 6

Move the limit under the radical sign.

$\sqrt{\frac{{(- 3)}^{lim_{x \to 8} x - 1 \cdot 1}}{\sqrt{lim_{x \to 8} x}}}$

Step 7

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\sqrt{\frac{{(- 3)}^{8 - 1 \cdot 1}}{\sqrt{lim_{x \to 8} x}}}$

Step 7.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\sqrt{\frac{{(- 3)}^{8 - 1 \cdot 1}}{\sqrt{8}}}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Multiply $- 1$ by $1$ .

$\sqrt{\frac{{(- 3)}^{8 - 1}}{\sqrt{8}}}$

Step 8.1.2

Subtract $1$ from $8$ .

$\sqrt{\frac{{(- 3)}^{7}}{\sqrt{8}}}$

Step 8.1.3

Raise $- 3$ to the power of $7$ .

$\sqrt{\frac{- 2187}{\sqrt{8}}}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 8.2.1.1

Factor $4$ out of $8$ .

$\sqrt{\frac{- 2187}{\sqrt{4 (2)}}}$

Step 8.2.1.2

Rewrite $4$ as $2^{2}$ .

$\sqrt{\frac{- 2187}{\sqrt{2^{2} \cdot 2}}}$

Step 8.2.2

Pull terms out from under the radical.

$\sqrt{\frac{- 2187}{2 \sqrt{2}}}$

Step 8.3

Move the negative in front of the fraction.

$\sqrt{- \frac{2187}{2 \sqrt{2}}}$

Step 8.4

Multiply $\frac{2187}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sqrt{- (\frac{2187}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}$

Step 8.5

Combine and simplify the denominator.

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Step 8.5.1

Multiply $\frac{2187}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sqrt{- \frac{2187 \sqrt{2}}{2 \sqrt{2} \sqrt{2}}}$

Step 8.5.2

Move $\sqrt{2}$ .

$\sqrt{- \frac{2187 \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}}$

Step 8.5.3

Raise $\sqrt{2}$ to the power of $1$ .

$\sqrt{- \frac{2187 \sqrt{2}}{2 ({\sqrt{2}}^{1} \sqrt{2})}}$

Step 8.5.4

Raise $\sqrt{2}$ to the power of $1$ .

$\sqrt{- \frac{2187 \sqrt{2}}{2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}}$

Step 8.5.5

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

$\sqrt{- \frac{2187 \sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}}$

Step 8.5.6

Add $1$ and $1$ .

$\sqrt{- \frac{2187 \sqrt{2}}{2 {\sqrt{2}}^{2}}}$

Step 8.5.7

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

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Step 8.5.7.1

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

$\sqrt{- \frac{2187 \sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}}}$

Step 8.5.7.2

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

$\sqrt{- \frac{2187 \sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}}}$

Step 8.5.7.3

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

$\sqrt{- \frac{2187 \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}}$

Step 8.5.7.4

Cancel the common factor of $2$ .

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Step 8.5.7.4.1

Cancel the common factor.

$\sqrt{- \frac{2187 \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}}$

Step 8.5.7.4.2

Rewrite the expression.

$\sqrt{- \frac{2187 \sqrt{2}}{2 \cdot 2^{1}}}$

Step 8.5.7.5

Evaluate the exponent.

$\sqrt{- \frac{2187 \sqrt{2}}{2 \cdot 2}}$

Step 8.6

Multiply $2$ by $2$ .

$\sqrt{- \frac{2187 \sqrt{2}}{4}}$

Step 8.7

Rewrite $- \frac{2187 \sqrt{2}}{4}$ as ${(\frac{27 i}{2})}^{2} \cdot (3 \sqrt{2})$ .

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Step 8.7.1

Rewrite $- 1$ as $i^{2}$ .

$\sqrt{i^{2} \frac{2187 \sqrt{2}}{4}}$

Step 8.7.2

Factor the perfect power $27^{2}$ out of $2187 \sqrt{2}$ .

$\sqrt{i^{2} \frac{27^{2} \cdot (3 \sqrt{2})}{4}}$

Step 8.7.3

Factor the perfect power $2^{2}$ out of $4$ .

$\sqrt{i^{2} \frac{27^{2} \cdot (3 \sqrt{2})}{2^{2} \cdot 1}}$

Step 8.7.4

Rearrange the fraction $\frac{27^{2} \cdot (3 \sqrt{2})}{2^{2} \cdot 1}$ .

$\sqrt{i^{2} ({(\frac{27}{2})}^{2} (3 \sqrt{2}))}$

Step 8.7.5

Rewrite $i^{2} {(\frac{27}{2})}^{2}$ as ${(\frac{27 i}{2})}^{2}$ .

$\sqrt{{(\frac{27 i}{2})}^{2} \cdot 3 \sqrt{2}}$

Step 8.7.6

Add parentheses.

$\sqrt{{(\frac{27 i}{2})}^{2} \cdot (3 \sqrt{2})}$

Step 8.8

Pull terms out from under the radical.

$\frac{27 i}{2} \sqrt{3 \sqrt{2}}$

Step 8.9

Combine $\frac{27 i}{2}$ and $\sqrt{3 \sqrt{2}}$ .

$\frac{27 i \sqrt{3 \sqrt{2}}}{2}$