Basic Math Examples

$\frac{\sqrt{7}}{\sqrt{11}} + \frac{\sqrt{11}}{\sqrt{7}}$

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{\sqrt{7}}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\frac{\sqrt{7}}{\sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}} + \frac{\sqrt{11}}{\sqrt{7}}$

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

Multiply $\frac{\sqrt{7}}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\frac{\sqrt{7} \sqrt{11}}{\sqrt{11} \sqrt{11}} + \frac{\sqrt{11}}{\sqrt{7}}$

Step 1.2.2

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

$\frac{\sqrt{7} \sqrt{11}}{{\sqrt{11}}^{1} \sqrt{11}} + \frac{\sqrt{11}}{\sqrt{7}}$

Step 1.2.3

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

$\frac{\sqrt{7} \sqrt{11}}{{\sqrt{11}}^{1} {\sqrt{11}}^{1}} + \frac{\sqrt{11}}{\sqrt{7}}$

Step 1.2.4

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

$\frac{\sqrt{7} \sqrt{11}}{{\sqrt{11}}^{1 + 1}} + \frac{\sqrt{11}}{\sqrt{7}}$

Step 1.2.5

Add $1$ and $1$ .

$\frac{\sqrt{7} \sqrt{11}}{{\sqrt{11}}^{2}} + \frac{\sqrt{11}}{\sqrt{7}}$

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

$\frac{\sqrt{7} \sqrt{11}}{{(11^{\frac{1}{2}})}^{2}} + \frac{\sqrt{11}}{\sqrt{7}}$

Step 1.2.6.2

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

$\frac{\sqrt{7} \sqrt{11}}{11^{\frac{1}{2} \cdot 2}} + \frac{\sqrt{11}}{\sqrt{7}}$

Step 1.2.6.3

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

$\frac{\sqrt{7} \sqrt{11}}{11^{\frac{2}{2}}} + \frac{\sqrt{11}}{\sqrt{7}}$

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.

$\frac{\sqrt{7} \sqrt{11}}{11^{\frac{2}{2}}} + \frac{\sqrt{11}}{\sqrt{7}}$

Step 1.2.6.4.2

Rewrite the expression.

$\frac{\sqrt{7} \sqrt{11}}{11^{1}} + \frac{\sqrt{11}}{\sqrt{7}}$

Step 1.2.6.5

Evaluate the exponent.

$\frac{\sqrt{7} \sqrt{11}}{11} + \frac{\sqrt{11}}{\sqrt{7}}$

Step 1.3

Simplify the numerator.

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Step 1.3.1

Combine using the product rule for radicals.

$\frac{\sqrt{7 \cdot 11}}{11} + \frac{\sqrt{11}}{\sqrt{7}}$

Step 1.3.2

Multiply $7$ by $11$ .

$\frac{\sqrt{77}}{11} + \frac{\sqrt{11}}{\sqrt{7}}$

Step 1.4

Multiply $\frac{\sqrt{11}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{\sqrt{77}}{11} + \frac{\sqrt{11}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 1.5

Combine and simplify the denominator.

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Step 1.5.1

Multiply $\frac{\sqrt{11}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{\sqrt{77}}{11} + \frac{\sqrt{11} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 1.5.2

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{\sqrt{77}}{11} + \frac{\sqrt{11} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Step 1.5.3

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{\sqrt{77}}{11} + \frac{\sqrt{11} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Step 1.5.4

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

$\frac{\sqrt{77}}{11} + \frac{\sqrt{11} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 1.5.5

Add $1$ and $1$ .

$\frac{\sqrt{77}}{11} + \frac{\sqrt{11} \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 1.5.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 1.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\frac{\sqrt{77}}{11} + \frac{\sqrt{11} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 1.5.6.2

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

$\frac{\sqrt{77}}{11} + \frac{\sqrt{11} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 1.5.6.3

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

$\frac{\sqrt{77}}{11} + \frac{\sqrt{11} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 1.5.6.4

Cancel the common factor of $2$ .

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Step 1.5.6.4.1

Cancel the common factor.

$\frac{\sqrt{77}}{11} + \frac{\sqrt{11} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 1.5.6.4.2

Rewrite the expression.

$\frac{\sqrt{77}}{11} + \frac{\sqrt{11} \sqrt{7}}{7^{1}}$

Step 1.5.6.5

Evaluate the exponent.

$\frac{\sqrt{77}}{11} + \frac{\sqrt{11} \sqrt{7}}{7}$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Combine using the product rule for radicals.

$\frac{\sqrt{77}}{11} + \frac{\sqrt{11 \cdot 7}}{7}$

Step 1.6.2

Multiply $11$ by $7$ .

$\frac{\sqrt{77}}{11} + \frac{\sqrt{77}}{7}$

Step 2

To write $\frac{\sqrt{77}}{11}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{\sqrt{77}}{11} \cdot \frac{7}{7} + \frac{\sqrt{77}}{7}$

Step 3

To write $\frac{\sqrt{77}}{7}$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$\frac{\sqrt{77}}{11} \cdot \frac{7}{7} + \frac{\sqrt{77}}{7} \cdot \frac{11}{11}$

Step 4

Write each expression with a common denominator of $77$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{\sqrt{77}}{11}$ by $\frac{7}{7}$ .

$\frac{\sqrt{77} \cdot 7}{11 \cdot 7} + \frac{\sqrt{77}}{7} \cdot \frac{11}{11}$

Step 4.2

Multiply $11$ by $7$ .

$\frac{\sqrt{77} \cdot 7}{77} + \frac{\sqrt{77}}{7} \cdot \frac{11}{11}$

Step 4.3

Multiply $\frac{\sqrt{77}}{7}$ by $\frac{11}{11}$ .

$\frac{\sqrt{77} \cdot 7}{77} + \frac{\sqrt{77} \cdot 11}{7 \cdot 11}$

Step 4.4

Multiply $7$ by $11$ .

$\frac{\sqrt{77} \cdot 7}{77} + \frac{\sqrt{77} \cdot 11}{77}$

Step 5

Combine the numerators over the common denominator.

$\frac{\sqrt{77} \cdot 7 + \sqrt{77} \cdot 11}{77}$

Step 6

Simplify the numerator.

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Step 6.1

Move $7$ to the left of $\sqrt{77}$ .

$\frac{7 \cdot \sqrt{77} + \sqrt{77} \cdot 11}{77}$

Step 6.2

Move $11$ to the left of $\sqrt{77}$ .

$\frac{7 \sqrt{77} + 11 \cdot \sqrt{77}}{77}$

Step 6.3

Add $7 \sqrt{77}$ and $11 \sqrt{77}$ .

$\frac{18 \sqrt{77}}{77}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{18 \sqrt{77}}{77}$

Decimal Form:

$2.05129037 \dots$