Calculus Examples

$lim_{h \to 0^{+}} \frac{\sqrt{h^{2} + 4 h + 11} - \sqrt{11}}{h}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{h \to 0^{+}} \sqrt{h^{2} + 4 h + 11} - \sqrt{11}}{lim_{h \to 0^{+}} h}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{lim_{h \to 0^{+}} \sqrt{h^{2} + 4 h + 11} - lim_{h \to 0^{+}} \sqrt{11}}{lim_{h \to 0^{+}} h}$

Step 1.1.2.2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{h \to 0^{+}} h^{2} + 4 h + 11} - lim_{h \to 0^{+}} \sqrt{11}}{lim_{h \to 0^{+}} h}$

Step 1.1.2.3

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{\sqrt{lim_{h \to 0^{+}} h^{2} + lim_{h \to 0^{+}} 4 h + lim_{h \to 0^{+}} 11} - lim_{h \to 0^{+}} \sqrt{11}}{lim_{h \to 0^{+}} h}$

Step 1.1.2.4

Move the exponent $2$ from $h^{2}$ outside the limit using the Limits Power Rule.

$\frac{\sqrt{{(lim_{h \to 0^{+}} h)}^{2} + lim_{h \to 0^{+}} 4 h + lim_{h \to 0^{+}} 11} - lim_{h \to 0^{+}} \sqrt{11}}{lim_{h \to 0^{+}} h}$

Step 1.1.2.5

Move the term $4$ outside of the limit because it is constant with respect to $h$ .

$\frac{\sqrt{{(lim_{h \to 0^{+}} h)}^{2} + 4 lim_{h \to 0^{+}} h + lim_{h \to 0^{+}} 11} - lim_{h \to 0^{+}} \sqrt{11}}{lim_{h \to 0^{+}} h}$

Step 1.1.2.6

Evaluate the limit of $11$ which is constant as $h$ approaches $0$ .

$\frac{\sqrt{{(lim_{h \to 0^{+}} h)}^{2} + 4 lim_{h \to 0^{+}} h + 11} - lim_{h \to 0^{+}} \sqrt{11}}{lim_{h \to 0^{+}} h}$

Step 1.1.2.7

Evaluate the limit of $\sqrt{11}$ which is constant as $h$ approaches $0$ .

$\frac{\sqrt{{(lim_{h \to 0^{+}} h)}^{2} + 4 lim_{h \to 0^{+}} h + 11} - \sqrt{11}}{lim_{h \to 0^{+}} h}$

Step 1.1.2.8

Evaluate the limits by plugging in $0$ for all occurrences of $h$ .

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Step 1.1.2.8.1

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{\sqrt{0^{2} + 4 lim_{h \to 0^{+}} h + 11} - \sqrt{11}}{lim_{h \to 0^{+}} h}$

Step 1.1.2.8.2

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{\sqrt{0^{2} + 4 \cdot 0 + 11} - \sqrt{11}}{lim_{h \to 0^{+}} h}$

Step 1.1.2.9

Simplify the answer.

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Step 1.1.2.9.1

Simplify each term.

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Step 1.1.2.9.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{\sqrt{0 + 4 \cdot 0 + 11} - \sqrt{11}}{lim_{h \to 0^{+}} h}$

Step 1.1.2.9.1.2

Multiply $4$ by $0$ .

$\frac{\sqrt{0 + 0 + 11} - \sqrt{11}}{lim_{h \to 0^{+}} h}$

Step 1.1.2.9.1.3

Add $0$ and $0$ .

$\frac{\sqrt{0 + 11} - \sqrt{11}}{lim_{h \to 0^{+}} h}$

Step 1.1.2.9.1.4

Add $0$ and $11$ .

$\frac{\sqrt{11} - \sqrt{11}}{lim_{h \to 0^{+}} h}$

Step 1.1.2.9.2

Subtract $\sqrt{11}$ from $\sqrt{11}$ .

$\frac{0}{lim_{h \to 0^{+}} h}$

Step 1.1.3

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{0}{0}$

Step 1.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{h \to 0^{+}} \frac{\sqrt{h^{2} + 4 h + 11} - \sqrt{11}}{h} = lim_{h \to 0^{+}} \frac{\frac{d}{d h} [\sqrt{h^{2} + 4 h + 11} - \sqrt{11}]}{\frac{d}{d h} [h]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{h \to 0^{+}} \frac{\frac{d}{d h} [\sqrt{h^{2} + 4 h + 11} - \sqrt{11}]}{\frac{d}{d h} [h]}$

Step 1.3.2

By the Sum Rule, the derivative of $\sqrt{h^{2} + 4 h + 11} - \sqrt{11}$ with respect to $h$ is $\frac{d}{d h} [\sqrt{h^{2} + 4 h + 11}] + \frac{d}{d h} [- \sqrt{11}]$ .

$lim_{h \to 0^{+}} \frac{\frac{d}{d h} [\sqrt{h^{2} + 4 h + 11}] + \frac{d}{d h} [- \sqrt{11}]}{\frac{d}{d h} [h]}$

Step 1.3.3

Evaluate $\frac{d}{d h} [\sqrt{h^{2} + 4 h + 11}]$ .

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Step 1.3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{h^{2} + 4 h + 11}$ as ${(h^{2} + 4 h + 11)}^{\frac{1}{2}}$ .

$lim_{h \to 0^{+}} \frac{\frac{d}{d h} [{(h^{2} + 4 h + 11)}^{\frac{1}{2}}] + \frac{d}{d h} [- \sqrt{11}]}{\frac{d}{d h} [h]}$

Step 1.3.3.2

Differentiate using the chain rule, which states that $\frac{d}{d h} [f (g (h))]$ is $f' (g (h)) g' (h)$ where $f (h) = h^{\frac{1}{2}}$ and $g (h) = h^{2} + 4 h + 11$ .

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Step 1.3.3.2.1

To apply the Chain Rule, set $u$ as $h^{2} + 4 h + 11$ .

$lim_{h \to 0^{+}} \frac{\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d h} [h^{2} + 4 h + 11] + \frac{d}{d h} [- \sqrt{11}]}{\frac{d}{d h} [h]}$

Step 1.3.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{2}$ .

$lim_{h \to 0^{+}} \frac{\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d h} [h^{2} + 4 h + 11] + \frac{d}{d h} [- \sqrt{11}]}{\frac{d}{d h} [h]}$

Step 1.3.3.2.3

Replace all occurrences of $u$ with $h^{2} + 4 h + 11$ .

$lim_{h \to 0^{+}} \frac{\frac{1}{2} {(h^{2} + 4 h + 11)}^{\frac{1}{2} - 1} \frac{d}{d h} [h^{2} + 4 h + 11] + \frac{d}{d h} [- \sqrt{11}]}{\frac{d}{d h} [h]}$

Step 1.3.3.3

By the Sum Rule, the derivative of $h^{2} + 4 h + 11$ with respect to $h$ is $\frac{d}{d h} [h^{2}] + \frac{d}{d h} [4 h] + \frac{d}{d h} [11]$ .

$lim_{h \to 0^{+}} \frac{\frac{1}{2} {(h^{2} + 4 h + 11)}^{\frac{1}{2} - 1} (\frac{d}{d h} [h^{2}] + \frac{d}{d h} [4 h] + \frac{d}{d h} [11]) + \frac{d}{d h} [- \sqrt{11}]}{\frac{d}{d h} [h]}$

Step 1.3.3.4

Differentiate using the Power Rule which states that $\frac{d}{d h} [h^{n}]$ is $n h^{n - 1}$ where $n = 2$ .

$lim_{h \to 0^{+}} \frac{\frac{1}{2} {(h^{2} + 4 h + 11)}^{\frac{1}{2} - 1} (2 h + \frac{d}{d h} [4 h] + \frac{d}{d h} [11]) + \frac{d}{d h} [- \sqrt{11}]}{\frac{d}{d h} [h]}$

Step 1.3.3.5

Since $4$ is constant with respect to $h$ , the derivative of $4 h$ with respect to $h$ is $4 \frac{d}{d h} [h]$ .

$lim_{h \to 0^{+}} \frac{\frac{1}{2} {(h^{2} + 4 h + 11)}^{\frac{1}{2} - 1} (2 h + 4 \frac{d}{d h} [h] + \frac{d}{d h} [11]) + \frac{d}{d h} [- \sqrt{11}]}{\frac{d}{d h} [h]}$

Step 1.3.3.6

Differentiate using the Power Rule which states that $\frac{d}{d h} [h^{n}]$ is $n h^{n - 1}$ where $n = 1$ .

$lim_{h \to 0^{+}} \frac{\frac{1}{2} {(h^{2} + 4 h + 11)}^{\frac{1}{2} - 1} (2 h + 4 \cdot 1 + \frac{d}{d h} [11]) + \frac{d}{d h} [- \sqrt{11}]}{\frac{d}{d h} [h]}$

Step 1.3.3.7

Since $11$ is constant with respect to $h$ , the derivative of $11$ with respect to $h$ is $0$ .

$lim_{h \to 0^{+}} \frac{\frac{1}{2} {(h^{2} + 4 h + 11)}^{\frac{1}{2} - 1} (2 h + 4 \cdot 1 + 0) + \frac{d}{d h} [- \sqrt{11}]}{\frac{d}{d h} [h]}$

Step 1.3.3.8

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{h \to 0^{+}} \frac{\frac{1}{2} {(h^{2} + 4 h + 11)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (2 h + 4 \cdot 1 + 0) + \frac{d}{d h} [- \sqrt{11}]}{\frac{d}{d h} [h]}$

Step 1.3.3.9

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

$lim_{h \to 0^{+}} \frac{\frac{1}{2} {(h^{2} + 4 h + 11)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (2 h + 4 \cdot 1 + 0) + \frac{d}{d h} [- \sqrt{11}]}{\frac{d}{d h} [h]}$

Step 1.3.3.10

Combine the numerators over the common denominator.

$lim_{h \to 0^{+}} \frac{\frac{1}{2} {(h^{2} + 4 h + 11)}^{\frac{1 - 1 \cdot 2}{2}} (2 h + 4 \cdot 1 + 0) + \frac{d}{d h} [- \sqrt{11}]}{\frac{d}{d h} [h]}$

Step 1.3.3.11

Simplify the numerator.

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Step 1.3.3.11.1

Multiply $- 1$ by $2$ .

$lim_{h \to 0^{+}} \frac{\frac{1}{2} {(h^{2} + 4 h + 11)}^{\frac{1 - 2}{2}} (2 h + 4 \cdot 1 + 0) + \frac{d}{d h} [- \sqrt{11}]}{\frac{d}{d h} [h]}$

Step 1.3.3.11.2

Subtract $2$ from $1$ .

$lim_{h \to 0^{+}} \frac{\frac{1}{2} {(h^{2} + 4 h + 11)}^{\frac{- 1}{2}} (2 h + 4 \cdot 1 + 0) + \frac{d}{d h} [- \sqrt{11}]}{\frac{d}{d h} [h]}$

Step 1.3.3.12

Move the negative in front of the fraction.

$lim_{h \to 0^{+}} \frac{\frac{1}{2} {(h^{2} + 4 h + 11)}^{- \frac{1}{2}} (2 h + 4 \cdot 1 + 0) + \frac{d}{d h} [- \sqrt{11}]}{\frac{d}{d h} [h]}$

Step 1.3.3.13

Multiply $4$ by $1$ .

$lim_{h \to 0^{+}} \frac{\frac{1}{2} {(h^{2} + 4 h + 11)}^{- \frac{1}{2}} (2 h + 4 + 0) + \frac{d}{d h} [- \sqrt{11}]}{\frac{d}{d h} [h]}$

Step 1.3.3.14

Add $2 h + 4$ and $0$ .

$lim_{h \to 0^{+}} \frac{\frac{1}{2} {(h^{2} + 4 h + 11)}^{- \frac{1}{2}} (2 h + 4) + \frac{d}{d h} [- \sqrt{11}]}{\frac{d}{d h} [h]}$

Step 1.3.3.15

Combine $\frac{1}{2}$ and ${(h^{2} + 4 h + 11)}^{- \frac{1}{2}}$ .

$lim_{h \to 0^{+}} \frac{\frac{{(h^{2} + 4 h + 11)}^{- \frac{1}{2}}}{2} (2 h + 4) + \frac{d}{d h} [- \sqrt{11}]}{\frac{d}{d h} [h]}$

Step 1.3.3.16

Move ${(h^{2} + 4 h + 11)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{h \to 0^{+}} \frac{\frac{1}{2 {(h^{2} + 4 h + 11)}^{\frac{1}{2}}} (2 h + 4) + \frac{d}{d h} [- \sqrt{11}]}{\frac{d}{d h} [h]}$

Step 1.3.4

Since $- \sqrt{11}$ is constant with respect to $h$ , the derivative of $- \sqrt{11}$ with respect to $h$ is $0$ .

$lim_{h \to 0^{+}} \frac{\frac{1}{2 {(h^{2} + 4 h + 11)}^{\frac{1}{2}}} (2 h + 4) + 0}{\frac{d}{d h} [h]}$

Step 1.3.5

Simplify.

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Step 1.3.5.1

Add $\frac{1}{2 {(h^{2} + 4 h + 11)}^{\frac{1}{2}}} (2 h + 4)$ and $0$ .

$lim_{h \to 0^{+}} \frac{\frac{1}{2 {(h^{2} + 4 h + 11)}^{\frac{1}{2}}} (2 h + 4)}{\frac{d}{d h} [h]}$

Step 1.3.5.2

Reorder the factors of $\frac{1}{2 {(h^{2} + 4 h + 11)}^{\frac{1}{2}}} (2 h + 4)$ .

$lim_{h \to 0^{+}} \frac{(2 h + 4) \frac{1}{2 {(h^{2} + 4 h + 11)}^{\frac{1}{2}}}}{\frac{d}{d h} [h]}$

Step 1.3.5.3

Multiply $2 h + 4$ by $\frac{1}{2 {(h^{2} + 4 h + 11)}^{\frac{1}{2}}}$ .

$lim_{h \to 0^{+}} \frac{\frac{2 h + 4}{2 {(h^{2} + 4 h + 11)}^{\frac{1}{2}}}}{\frac{d}{d h} [h]}$

Step 1.3.5.4

Factor $2$ out of $2 h$ .

$lim_{h \to 0^{+}} \frac{\frac{2 (h) + 4}{2 {(h^{2} + 4 h + 11)}^{\frac{1}{2}}}}{\frac{d}{d h} [h]}$

Step 1.3.5.5

Factor $2$ out of $4$ .

$lim_{h \to 0^{+}} \frac{\frac{2 (h) + 2 \cdot 2}{2 {(h^{2} + 4 h + 11)}^{\frac{1}{2}}}}{\frac{d}{d h} [h]}$

Step 1.3.5.6

Factor $2$ out of $2 (h) + 2 (2)$ .

$lim_{h \to 0^{+}} \frac{\frac{2 (h + 2)}{2 {(h^{2} + 4 h + 11)}^{\frac{1}{2}}}}{\frac{d}{d h} [h]}$

Step 1.3.5.7

Cancel the common factors.

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Step 1.3.5.7.1

Factor $2$ out of $2 {(h^{2} + 4 h + 11)}^{\frac{1}{2}}$ .

$lim_{h \to 0^{+}} \frac{\frac{2 (h + 2)}{2 ({(h^{2} + 4 h + 11)}^{\frac{1}{2}})}}{\frac{d}{d h} [h]}$

Step 1.3.5.7.2

Cancel the common factor.

$lim_{h \to 0^{+}} \frac{\frac{2 (h + 2)}{2 {(h^{2} + 4 h + 11)}^{\frac{1}{2}}}}{\frac{d}{d h} [h]}$

Step 1.3.5.7.3

Rewrite the expression.

$lim_{h \to 0^{+}} \frac{\frac{h + 2}{{(h^{2} + 4 h + 11)}^{\frac{1}{2}}}}{\frac{d}{d h} [h]}$

Step 1.3.6

Differentiate using the Power Rule which states that $\frac{d}{d h} [h^{n}]$ is $n h^{n - 1}$ where $n = 1$ .

$lim_{h \to 0^{+}} \frac{\frac{h + 2}{{(h^{2} + 4 h + 11)}^{\frac{1}{2}}}}{1}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{h \to 0^{+}} \frac{h + 2}{{(h^{2} + 4 h + 11)}^{\frac{1}{2}}} \cdot 1$

Step 1.5

Rewrite ${(h^{2} + 4 h + 11)}^{\frac{1}{2}}$ as $\sqrt{h^{2} + 4 h + 11}$ .

$lim_{h \to 0^{+}} \frac{h + 2}{\sqrt{h^{2} + 4 h + 11}} \cdot 1$

Step 1.6

Multiply $\frac{h + 2}{\sqrt{h^{2} + 4 h + 11}}$ by $1$ .

$lim_{h \to 0^{+}} \frac{h + 2}{\sqrt{h^{2} + 4 h + 11}}$

Step 2

Evaluate the limit.

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Step 2.1

Split the limit using the Limits Quotient Rule on the limit as $h$ approaches $0$ .

$\frac{lim_{h \to 0^{+}} h + 2}{lim_{h \to 0^{+}} \sqrt{h^{2} + 4 h + 11}}$

Step 2.2

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{lim_{h \to 0^{+}} h + lim_{h \to 0^{+}} 2}{lim_{h \to 0^{+}} \sqrt{h^{2} + 4 h + 11}}$

Step 2.3

Evaluate the limit of $2$ which is constant as $h$ approaches $0$ .

$\frac{lim_{h \to 0^{+}} h + 2}{lim_{h \to 0^{+}} \sqrt{h^{2} + 4 h + 11}}$

Step 2.4

Move the limit under the radical sign.

$\frac{lim_{h \to 0^{+}} h + 2}{\sqrt{lim_{h \to 0^{+}} h^{2} + 4 h + 11}}$

Step 2.5

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{lim_{h \to 0^{+}} h + 2}{\sqrt{lim_{h \to 0^{+}} h^{2} + lim_{h \to 0^{+}} 4 h + lim_{h \to 0^{+}} 11}}$

Step 2.6

Move the exponent $2$ from $h^{2}$ outside the limit using the Limits Power Rule.

$\frac{lim_{h \to 0^{+}} h + 2}{\sqrt{{(lim_{h \to 0^{+}} h)}^{2} + lim_{h \to 0^{+}} 4 h + lim_{h \to 0^{+}} 11}}$

Step 2.7

Move the term $4$ outside of the limit because it is constant with respect to $h$ .

$\frac{lim_{h \to 0^{+}} h + 2}{\sqrt{{(lim_{h \to 0^{+}} h)}^{2} + 4 lim_{h \to 0^{+}} h + lim_{h \to 0^{+}} 11}}$

Step 2.8

Evaluate the limit of $11$ which is constant as $h$ approaches $0$ .

$\frac{lim_{h \to 0^{+}} h + 2}{\sqrt{{(lim_{h \to 0^{+}} h)}^{2} + 4 lim_{h \to 0^{+}} h + 11}}$

Step 3

Evaluate the limits by plugging in $0$ for all occurrences of $h$ .

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Step 3.1

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{0 + 2}{\sqrt{{(lim_{h \to 0^{+}} h)}^{2} + 4 lim_{h \to 0^{+}} h + 11}}$

Step 3.2

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{0 + 2}{\sqrt{0^{2} + 4 lim_{h \to 0^{+}} h + 11}}$

Step 3.3

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{0 + 2}{\sqrt{0^{2} + 4 \cdot 0 + 11}}$

Step 4

Simplify the answer.

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Step 4.1

Add $0$ and $2$ .

$\frac{2}{\sqrt{0^{2} + 4 \cdot 0 + 11}}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Raising $0$ to any positive power yields $0$ .

$\frac{2}{\sqrt{0 + 4 \cdot 0 + 11}}$

Step 4.2.2

Multiply $4$ by $0$ .

$\frac{2}{\sqrt{0 + 0 + 11}}$

Step 4.2.3

Add $0$ and $0$ .

$\frac{2}{\sqrt{0 + 11}}$

Step 4.2.4

Add $0$ and $11$ .

$\frac{2}{\sqrt{11}}$

Step 4.3

Multiply $\frac{2}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\frac{2}{\sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}}$

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

Multiply $\frac{2}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\frac{2 \sqrt{11}}{\sqrt{11} \sqrt{11}}$

Step 4.4.2

Raise $\sqrt{11}$ to the power of $1$ .

$\frac{2 \sqrt{11}}{{\sqrt{11}}^{1} \sqrt{11}}$

Step 4.4.3

Raise $\sqrt{11}$ to the power of $1$ .

$\frac{2 \sqrt{11}}{{\sqrt{11}}^{1} {\sqrt{11}}^{1}}$

Step 4.4.4

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

$\frac{2 \sqrt{11}}{{\sqrt{11}}^{1 + 1}}$

Step 4.4.5

Add $1$ and $1$ .

$\frac{2 \sqrt{11}}{{\sqrt{11}}^{2}}$

Step 4.4.6

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\frac{2 \sqrt{11}}{{(11^{\frac{1}{2}})}^{2}}$

Step 4.4.6.2

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

$\frac{2 \sqrt{11}}{11^{\frac{1}{2} \cdot 2}}$

Step 4.4.6.3

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

$\frac{2 \sqrt{11}}{11^{\frac{2}{2}}}$

Step 4.4.6.4

Cancel the common factor of $2$ .

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Step 4.4.6.4.1

Cancel the common factor.

$\frac{2 \sqrt{11}}{11^{\frac{2}{2}}}$

Step 4.4.6.4.2

Rewrite the expression.

$\frac{2 \sqrt{11}}{11^{1}}$

Step 4.4.6.5

Evaluate the exponent.

$\frac{2 \sqrt{11}}{11}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{11}}{11}$

Decimal Form:

$0.60302268 \dots$