Simplify ( square root of 7)/( square root of 11)+( square root of 11)/( square root of 7)

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

Step 1

Simplify each term.

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $7$ by $11$ .

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Simplify the numerator.

Combine using the product rule for radicals.

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

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 $1$ .

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

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

Multiply $11$ by $7$ .

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

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

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

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.

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

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

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

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

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:

$2.05129037 \dots$

Simplify ( square root of 7)/( square root of 11)+( square root of 11)/( square root of 7)

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