Simplify ( square root of 11-5)(5- square root of 11)

$(\sqrt{11} - 5) (5 - \sqrt{11})$

Step 1

Expand $(\sqrt{11} - 5) (5 - \sqrt{11})$ using the FOIL Method.

Apply the distributive property.

$\sqrt{11} (5 - \sqrt{11}) - 5 (5 - \sqrt{11})$

Apply the distributive property.

$\sqrt{11} \cdot 5 + \sqrt{11} (- \sqrt{11}) - 5 (5 - \sqrt{11})$

Apply the distributive property.

$\sqrt{11} \cdot 5 + \sqrt{11} (- \sqrt{11}) - 5 \cdot 5 - 5 (- \sqrt{11})$

Step 2

Simplify and combine like terms.

Simplify each term.

Move $5$ to the left of $\sqrt{11}$ .

$5 \cdot \sqrt{11} + \sqrt{11} (- \sqrt{11}) - 5 \cdot 5 - 5 (- \sqrt{11})$

Multiply $\sqrt{11} (- \sqrt{11})$ .

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

$5 \sqrt{11} - ({\sqrt{11}}^{1} \sqrt{11}) - 5 \cdot 5 - 5 (- \sqrt{11})$

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

$5 \sqrt{11} - ({\sqrt{11}}^{1} {\sqrt{11}}^{1}) - 5 \cdot 5 - 5 (- \sqrt{11})$

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

$5 \sqrt{11} - {\sqrt{11}}^{1 + 1} - 5 \cdot 5 - 5 (- \sqrt{11})$

Add $1$ and $1$ .

$5 \sqrt{11} - {\sqrt{11}}^{2} - 5 \cdot 5 - 5 (- \sqrt{11})$

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

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

$5 \sqrt{11} - {(11^{\frac{1}{2}})}^{2} - 5 \cdot 5 - 5 (- \sqrt{11})$

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

$5 \sqrt{11} - 11^{\frac{1}{2} \cdot 2} - 5 \cdot 5 - 5 (- \sqrt{11})$

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

$5 \sqrt{11} - 11^{\frac{2}{2}} - 5 \cdot 5 - 5 (- \sqrt{11})$

Cancel the common factor of $2$ .

Cancel the common factor.

$5 \sqrt{11} - 11^{\frac{2}{2}} - 5 \cdot 5 - 5 (- \sqrt{11})$

Rewrite the expression.

$5 \sqrt{11} - 11^{1} - 5 \cdot 5 - 5 (- \sqrt{11})$

Evaluate the exponent.

$5 \sqrt{11} - 1 \cdot 11 - 5 \cdot 5 - 5 (- \sqrt{11})$

Multiply $- 1$ by $11$ .

$5 \sqrt{11} - 11 - 5 \cdot 5 - 5 (- \sqrt{11})$

Multiply $- 5$ by $5$ .

$5 \sqrt{11} - 11 - 25 - 5 (- \sqrt{11})$

Multiply $- 1$ by $- 5$ .

$5 \sqrt{11} - 11 - 25 + 5 \sqrt{11}$

Add $5 \sqrt{11}$ and $5 \sqrt{11}$ .

$- 11 - 25 + 10 \sqrt{11}$

Subtract $25$ from $- 11$ .

$- 36 + 10 \sqrt{11}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- 36 + 10 \sqrt{11}$

Decimal Form:

$- 2.83375209 \dots$

Simplify ( square root of 11-5)(5- square root of 11)

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