Evaluate square root of (( square root of 11)/11)^2+(( square root of 11)/11)^2+((3 square root of 11)/11)^2

$\sqrt{{(\frac{\sqrt{11}}{11})}^{2} + {(\frac{\sqrt{11}}{11})}^{2} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Apply the product rule to $\frac{\sqrt{11}}{11}$ .

$\sqrt{\frac{{\sqrt{11}}^{2}}{11^{2}} + {(\frac{\sqrt{11}}{11})}^{2} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

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

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

$\sqrt{\frac{{(11^{\frac{1}{2}})}^{2}}{11^{2}} + {(\frac{\sqrt{11}}{11})}^{2} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

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

$\sqrt{\frac{11^{\frac{1}{2} \cdot 2}}{11^{2}} + {(\frac{\sqrt{11}}{11})}^{2} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

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

$\sqrt{\frac{11^{\frac{2}{2}}}{11^{2}} + {(\frac{\sqrt{11}}{11})}^{2} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{\frac{11^{\frac{2}{2}}}{11^{2}} + {(\frac{\sqrt{11}}{11})}^{2} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Rewrite the expression.

$\sqrt{\frac{11^{1}}{11^{2}} + {(\frac{\sqrt{11}}{11})}^{2} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Evaluate the exponent.

$\sqrt{\frac{11}{11^{2}} + {(\frac{\sqrt{11}}{11})}^{2} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Raise $11$ to the power of $2$ .

$\sqrt{\frac{11}{121} + {(\frac{\sqrt{11}}{11})}^{2} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Cancel the common factor of $11$ and $121$ .

Factor $11$ out of $11$ .

$\sqrt{\frac{11 (1)}{121} + {(\frac{\sqrt{11}}{11})}^{2} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Cancel the common factors.

Factor $11$ out of $121$ .

$\sqrt{\frac{11 \cdot 1}{11 \cdot 11} + {(\frac{\sqrt{11}}{11})}^{2} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Cancel the common factor.

$\sqrt{\frac{11 \cdot 1}{11 \cdot 11} + {(\frac{\sqrt{11}}{11})}^{2} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Rewrite the expression.

$\sqrt{\frac{1}{11} + {(\frac{\sqrt{11}}{11})}^{2} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Apply the product rule to $\frac{\sqrt{11}}{11}$ .

$\sqrt{\frac{1}{11} + \frac{{\sqrt{11}}^{2}}{11^{2}} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

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

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

$\sqrt{\frac{1}{11} + \frac{{(11^{\frac{1}{2}})}^{2}}{11^{2}} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

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

$\sqrt{\frac{1}{11} + \frac{11^{\frac{1}{2} \cdot 2}}{11^{2}} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

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

$\sqrt{\frac{1}{11} + \frac{11^{\frac{2}{2}}}{11^{2}} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{\frac{1}{11} + \frac{11^{\frac{2}{2}}}{11^{2}} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Rewrite the expression.

$\sqrt{\frac{1}{11} + \frac{11^{1}}{11^{2}} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Evaluate the exponent.

$\sqrt{\frac{1}{11} + \frac{11}{11^{2}} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Raise $11$ to the power of $2$ .

$\sqrt{\frac{1}{11} + \frac{11}{121} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Cancel the common factor of $11$ and $121$ .

Factor $11$ out of $11$ .

$\sqrt{\frac{1}{11} + \frac{11 (1)}{121} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Cancel the common factors.

Factor $11$ out of $121$ .

$\sqrt{\frac{1}{11} + \frac{11 \cdot 1}{11 \cdot 11} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Cancel the common factor.

$\sqrt{\frac{1}{11} + \frac{11 \cdot 1}{11 \cdot 11} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Rewrite the expression.

$\sqrt{\frac{1}{11} + \frac{1}{11} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{3 \sqrt{11}}{11}$ .

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{{(3 \sqrt{11})}^{2}}{11^{2}}}$

Apply the product rule to $3 \sqrt{11}$ .

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{3^{2} {\sqrt{11}}^{2}}{11^{2}}}$

Simplify the numerator.

Raise $3$ to the power of $2$ .

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{9 {\sqrt{11}}^{2}}{11^{2}}}$

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

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

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{9 {(11^{\frac{1}{2}})}^{2}}{11^{2}}}$

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

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{9 \cdot 11^{\frac{1}{2} \cdot 2}}{11^{2}}}$

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

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{9 \cdot 11^{\frac{2}{2}}}{11^{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{9 \cdot 11^{\frac{2}{2}}}{11^{2}}}$

Rewrite the expression.

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{9 \cdot 11^{1}}{11^{2}}}$

Evaluate the exponent.

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{9 \cdot 11}{11^{2}}}$

Reduce the expression by cancelling the common factors.

Raise $11$ to the power of $2$ .

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{9 \cdot 11}{121}}$

Multiply $9$ by $11$ .

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{99}{121}}$

Cancel the common factor of $99$ and $121$ .

Factor $11$ out of $99$ .

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{11 (9)}{121}}$

Cancel the common factors.

Factor $11$ out of $121$ .

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{11 \cdot 9}{11 \cdot 11}}$

Cancel the common factor.

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{11 \cdot 9}{11 \cdot 11}}$

Rewrite the expression.

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{9}{11}}$

Simplify the expression.

Combine the numerators over the common denominator.

$\sqrt{\frac{1 + 1}{11} + \frac{9}{11}}$

Add $1$ and $1$ .

$\sqrt{\frac{2}{11} + \frac{9}{11}}$

Combine the numerators over the common denominator.

$\sqrt{\frac{2 + 9}{11}}$

Add $2$ and $9$ .

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

Divide $11$ by $11$ .

$\sqrt{1}$

Any root of $1$ is $1$ .

$1$

Evaluate square root of (( square root of 11)/11)^2+(( square root of 11)/11)^2+((3 square root of 11)/11)^2

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