Basic Math Examples

$\frac{1}{\sqrt{11}} + \frac{2}{\sqrt{1331}} + \frac{1}{\sqrt{161.051}}$

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{1}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\frac{1}{\sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}} + \frac{2}{\sqrt{1331}} + \frac{1}{\sqrt{161.051}}$

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

Multiply $\frac{1}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\frac{\sqrt{11}}{\sqrt{11} \sqrt{11}} + \frac{2}{\sqrt{1331}} + \frac{1}{\sqrt{161.051}}$

Step 1.2.2

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

$\frac{\sqrt{11}}{{\sqrt{11}}^{1} \sqrt{11}} + \frac{2}{\sqrt{1331}} + \frac{1}{\sqrt{161.051}}$

Step 1.2.3

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

$\frac{\sqrt{11}}{{\sqrt{11}}^{1} {\sqrt{11}}^{1}} + \frac{2}{\sqrt{1331}} + \frac{1}{\sqrt{161.051}}$

Step 1.2.4

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

$\frac{\sqrt{11}}{{\sqrt{11}}^{1 + 1}} + \frac{2}{\sqrt{1331}} + \frac{1}{\sqrt{161.051}}$

Step 1.2.5

Add $1$ and $1$ .

$\frac{\sqrt{11}}{{\sqrt{11}}^{2}} + \frac{2}{\sqrt{1331}} + \frac{1}{\sqrt{161.051}}$

Step 1.2.6

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

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Step 1.2.6.1

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

$\frac{\sqrt{11}}{{(11^{\frac{1}{2}})}^{2}} + \frac{2}{\sqrt{1331}} + \frac{1}{\sqrt{161.051}}$

Step 1.2.6.2

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

$\frac{\sqrt{11}}{11^{\frac{1}{2} \cdot 2}} + \frac{2}{\sqrt{1331}} + \frac{1}{\sqrt{161.051}}$

Step 1.2.6.3

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

$\frac{\sqrt{11}}{11^{\frac{2}{2}}} + \frac{2}{\sqrt{1331}} + \frac{1}{\sqrt{161.051}}$

Step 1.2.6.4

Cancel the common factor of $2$ .

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Step 1.2.6.4.1

Cancel the common factor.

$\frac{\sqrt{11}}{11^{\frac{2}{2}}} + \frac{2}{\sqrt{1331}} + \frac{1}{\sqrt{161.051}}$

Step 1.2.6.4.2

Rewrite the expression.

$\frac{\sqrt{11}}{11^{1}} + \frac{2}{\sqrt{1331}} + \frac{1}{\sqrt{161.051}}$

Step 1.2.6.5

Evaluate the exponent.

$\frac{\sqrt{11}}{11} + \frac{2}{\sqrt{1331}} + \frac{1}{\sqrt{161.051}}$

Step 1.3

Simplify the denominator.

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Step 1.3.1

Rewrite $1331$ as $11^{2} \cdot 11$ .

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Step 1.3.1.1

Factor $121$ out of $1331$ .

$\frac{\sqrt{11}}{11} + \frac{2}{\sqrt{121 (11)}} + \frac{1}{\sqrt{161.051}}$

Step 1.3.1.2

Rewrite $121$ as $11^{2}$ .

$\frac{\sqrt{11}}{11} + \frac{2}{\sqrt{11^{2} \cdot 11}} + \frac{1}{\sqrt{161.051}}$

Step 1.3.2

Pull terms out from under the radical.

$\frac{\sqrt{11}}{11} + \frac{2}{11 \sqrt{11}} + \frac{1}{\sqrt{161.051}}$

Step 1.4

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

$\frac{\sqrt{11}}{11} + \frac{2}{11 \sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}} + \frac{1}{\sqrt{161.051}}$

Step 1.5

Combine and simplify the denominator.

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Step 1.5.1

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

$\frac{\sqrt{11}}{11} + \frac{2 \sqrt{11}}{11 \sqrt{11} \sqrt{11}} + \frac{1}{\sqrt{161.051}}$

Step 1.5.2

Move $\sqrt{11}$ .

$\frac{\sqrt{11}}{11} + \frac{2 \sqrt{11}}{11 (\sqrt{11} \sqrt{11})} + \frac{1}{\sqrt{161.051}}$

Step 1.5.3

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

$\frac{\sqrt{11}}{11} + \frac{2 \sqrt{11}}{11 ({\sqrt{11}}^{1} \sqrt{11})} + \frac{1}{\sqrt{161.051}}$

Step 1.5.4

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

$\frac{\sqrt{11}}{11} + \frac{2 \sqrt{11}}{11 ({\sqrt{11}}^{1} {\sqrt{11}}^{1})} + \frac{1}{\sqrt{161.051}}$

Step 1.5.5

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

$\frac{\sqrt{11}}{11} + \frac{2 \sqrt{11}}{11 {\sqrt{11}}^{1 + 1}} + \frac{1}{\sqrt{161.051}}$

Step 1.5.6

Add $1$ and $1$ .

$\frac{\sqrt{11}}{11} + \frac{2 \sqrt{11}}{11 {\sqrt{11}}^{2}} + \frac{1}{\sqrt{161.051}}$

Step 1.5.7

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

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Step 1.5.7.1

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

$\frac{\sqrt{11}}{11} + \frac{2 \sqrt{11}}{11 {(11^{\frac{1}{2}})}^{2}} + \frac{1}{\sqrt{161.051}}$

Step 1.5.7.2

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

$\frac{\sqrt{11}}{11} + \frac{2 \sqrt{11}}{11 \cdot 11^{\frac{1}{2} \cdot 2}} + \frac{1}{\sqrt{161.051}}$

Step 1.5.7.3

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

$\frac{\sqrt{11}}{11} + \frac{2 \sqrt{11}}{11 \cdot 11^{\frac{2}{2}}} + \frac{1}{\sqrt{161.051}}$

Step 1.5.7.4

Cancel the common factor of $2$ .

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Step 1.5.7.4.1

Cancel the common factor.

$\frac{\sqrt{11}}{11} + \frac{2 \sqrt{11}}{11 \cdot 11^{\frac{2}{2}}} + \frac{1}{\sqrt{161.051}}$

Step 1.5.7.4.2

Rewrite the expression.

$\frac{\sqrt{11}}{11} + \frac{2 \sqrt{11}}{11 \cdot 11^{1}} + \frac{1}{\sqrt{161.051}}$

Step 1.5.7.5

Evaluate the exponent.

$\frac{\sqrt{11}}{11} + \frac{2 \sqrt{11}}{11 \cdot 11} + \frac{1}{\sqrt{161.051}}$

Step 1.6

Multiply $11$ by $11$ .

$\frac{\sqrt{11}}{11} + \frac{2 \sqrt{11}}{121} + \frac{1}{\sqrt{161.051}}$

Step 1.7

Multiply $\frac{1}{\sqrt{161.051}}$ by $\frac{\sqrt{161.051}}{\sqrt{161.051}}$ .

$\frac{\sqrt{11}}{11} + \frac{2 \sqrt{11}}{121} + \frac{1}{\sqrt{161.051}} \cdot \frac{\sqrt{161.051}}{\sqrt{161.051}}$

Step 1.8

Combine and simplify the denominator.

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Step 1.8.1

Multiply $\frac{1}{\sqrt{161.051}}$ by $\frac{\sqrt{161.051}}{\sqrt{161.051}}$ .

$\frac{\sqrt{11}}{11} + \frac{2 \sqrt{11}}{121} + \frac{\sqrt{161.051}}{\sqrt{161.051} \sqrt{161.051}}$

Step 1.8.2

Raise $\sqrt{161.051}$ to the power of $1$ .

$\frac{\sqrt{11}}{11} + \frac{2 \sqrt{11}}{121} + \frac{\sqrt{161.051}}{{\sqrt{161.051}}^{1} \sqrt{161.051}}$

Step 1.8.3

Raise $\sqrt{161.051}$ to the power of $1$ .

$\frac{\sqrt{11}}{11} + \frac{2 \sqrt{11}}{121} + \frac{\sqrt{161.051}}{{\sqrt{161.051}}^{1} {\sqrt{161.051}}^{1}}$

Step 1.8.4

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

$\frac{\sqrt{11}}{11} + \frac{2 \sqrt{11}}{121} + \frac{\sqrt{161.051}}{{\sqrt{161.051}}^{1 + 1}}$

Step 1.8.5

Add $1$ and $1$ .

$\frac{\sqrt{11}}{11} + \frac{2 \sqrt{11}}{121} + \frac{\sqrt{161.051}}{{\sqrt{161.051}}^{2}}$

Step 1.8.6

Rewrite ${\sqrt{161.051}}^{2}$ as $161.051$ .

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Step 1.8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{161.051}$ as ${161.051}^{\frac{1}{2}}$ .

$\frac{\sqrt{11}}{11} + \frac{2 \sqrt{11}}{121} + \frac{\sqrt{161.051}}{{({161.051}^{\frac{1}{2}})}^{2}}$

Step 1.8.6.2

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

$\frac{\sqrt{11}}{11} + \frac{2 \sqrt{11}}{121} + \frac{\sqrt{161.051}}{{161.051}^{\frac{1}{2} \cdot 2}}$

Step 1.8.6.3

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

$\frac{\sqrt{11}}{11} + \frac{2 \sqrt{11}}{121} + \frac{\sqrt{161.051}}{{161.051}^{\frac{2}{2}}}$

Step 1.8.6.4

Cancel the common factor of $2$ .

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Step 1.8.6.4.1

Cancel the common factor.

$\frac{\sqrt{11}}{11} + \frac{2 \sqrt{11}}{121} + \frac{\sqrt{161.051}}{{161.051}^{\frac{2}{2}}}$

Step 1.8.6.4.2

Rewrite the expression.

$\frac{\sqrt{11}}{11} + \frac{2 \sqrt{11}}{121} + \frac{\sqrt{161.051}}{{161.051}^{1}}$

Step 1.8.6.5

Evaluate the exponent.

$\frac{\sqrt{11}}{11} + \frac{2 \sqrt{11}}{121} + \frac{\sqrt{161.051}}{161.051}$

Step 1.9

Evaluate the root.

$\frac{\sqrt{11}}{11} + \frac{2 \sqrt{11}}{121} + \frac{12.69058706}{161.051}$

Step 1.10

Divide $12.69058706$ by $161.051$ .

$\frac{\sqrt{11}}{11} + \frac{2 \sqrt{11}}{121} + 0.07879856$

Step 2

Find the common denominator.

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Step 2.1

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

$\frac{\sqrt{11}}{11} \cdot \frac{11}{11} + \frac{2 \sqrt{11}}{121} + 0.07879856$

Step 2.2

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

$\frac{\sqrt{11} \cdot 11}{11 \cdot 11} + \frac{2 \sqrt{11}}{121} + 0.07879856$

Step 2.3

Write $0.07879856$ as a fraction with denominator $1$ .

$\frac{\sqrt{11} \cdot 11}{11 \cdot 11} + \frac{2 \sqrt{11}}{121} + \frac{0.07879856}{1}$

Step 2.4

Multiply $\frac{0.07879856}{1}$ by $\frac{121}{121}$ .

$\frac{\sqrt{11} \cdot 11}{11 \cdot 11} + \frac{2 \sqrt{11}}{121} + \frac{0.07879856}{1} \cdot \frac{121}{121}$

Step 2.5

Multiply $\frac{0.07879856}{1}$ by $\frac{121}{121}$ .

$\frac{\sqrt{11} \cdot 11}{11 \cdot 11} + \frac{2 \sqrt{11}}{121} + \frac{0.07879856 \cdot 121}{121}$

Step 2.6

Multiply $11$ by $11$ .

$\frac{\sqrt{11} \cdot 11}{121} + \frac{2 \sqrt{11}}{121} + \frac{0.07879856 \cdot 121}{121}$

Step 3

Combine the numerators over the common denominator.

$\frac{\sqrt{11} \cdot 11 + 2 \sqrt{11} + 0.07879856 \cdot 121}{121}$

Step 4

Simplify each term.

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Step 4.1

Move $11$ to the left of $\sqrt{11}$ .

$\frac{11 \cdot \sqrt{11} + 2 \sqrt{11} + 0.07879856 \cdot 121}{121}$

Step 4.2

Multiply $0.07879856$ by $121$ .

$\frac{11 \sqrt{11} + 2 \sqrt{11} + 9.53462589}{121}$

Step 5

Add $11 \sqrt{11}$ and $2 \sqrt{11}$ .

$\frac{13 \sqrt{11} + 9.53462589}{121}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{13 \sqrt{11} + 9.53462589}{121}$

Decimal Form:

$0.43513015 \dots$