Evaluate (7- square root of 2)/(7+ square root of 2)

$\frac{7 - \sqrt{2}}{7 + \sqrt{2}}$

Step 1

Multiply $\frac{7 - \sqrt{2}}{7 + \sqrt{2}}$ by $\frac{7 - \sqrt{2}}{7 - \sqrt{2}}$ .

$\frac{7 - \sqrt{2}}{7 + \sqrt{2}} \cdot \frac{7 - \sqrt{2}}{7 - \sqrt{2}}$

Step 2

Combine fractions.

Multiply $\frac{7 - \sqrt{2}}{7 + \sqrt{2}}$ by $\frac{7 - \sqrt{2}}{7 - \sqrt{2}}$ .

$\frac{(7 - \sqrt{2}) (7 - \sqrt{2})}{(7 + \sqrt{2}) (7 - \sqrt{2})}$

Expand the denominator using the FOIL method.

$\frac{(7 - \sqrt{2}) (7 - \sqrt{2})}{49 - 7 \sqrt{2} + \sqrt{2} \cdot 7 - {\sqrt{2}}^{2}}$

Simplify.

$\frac{(7 - \sqrt{2}) (7 - \sqrt{2})}{47}$

Step 3

Simplify the numerator.

Raise $7 - \sqrt{2}$ to the power of $1$ .

$\frac{{(7 - \sqrt{2})}^{1} (7 - \sqrt{2})}{47}$

Raise $7 - \sqrt{2}$ to the power of $1$ .

$\frac{{(7 - \sqrt{2})}^{1} {(7 - \sqrt{2})}^{1}}{47}$

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

$\frac{{(7 - \sqrt{2})}^{1 + 1}}{47}$

Add $1$ and $1$ .

$\frac{{(7 - \sqrt{2})}^{2}}{47}$

Step 4

Rewrite ${(7 - \sqrt{2})}^{2}$ as $(7 - \sqrt{2}) (7 - \sqrt{2})$ .

$\frac{(7 - \sqrt{2}) (7 - \sqrt{2})}{47}$

Step 5

Expand $(7 - \sqrt{2}) (7 - \sqrt{2})$ using the FOIL Method.

Apply the distributive property.

$\frac{7 (7 - \sqrt{2}) - \sqrt{2} (7 - \sqrt{2})}{47}$

Apply the distributive property.

$\frac{7 \cdot 7 + 7 (- \sqrt{2}) - \sqrt{2} (7 - \sqrt{2})}{47}$

Apply the distributive property.

$\frac{7 \cdot 7 + 7 (- \sqrt{2}) - \sqrt{2} \cdot 7 - \sqrt{2} (- \sqrt{2})}{47}$

Step 6

Simplify and combine like terms.

Simplify each term.

Multiply $7$ by $7$ .

$\frac{49 + 7 (- \sqrt{2}) - \sqrt{2} \cdot 7 - \sqrt{2} (- \sqrt{2})}{47}$

Multiply $- 1$ by $7$ .

$\frac{49 - 7 \sqrt{2} - \sqrt{2} \cdot 7 - \sqrt{2} (- \sqrt{2})}{47}$

Multiply $7$ by $- 1$ .

$\frac{49 - 7 \sqrt{2} - 7 \sqrt{2} - \sqrt{2} (- \sqrt{2})}{47}$

Multiply $- \sqrt{2} (- \sqrt{2})$ .

Multiply $- 1$ by $- 1$ .

$\frac{49 - 7 \sqrt{2} - 7 \sqrt{2} + 1 \sqrt{2} \sqrt{2}}{47}$

Multiply $\sqrt{2}$ by $1$ .

$\frac{49 - 7 \sqrt{2} - 7 \sqrt{2} + \sqrt{2} \sqrt{2}}{47}$

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{49 - 7 \sqrt{2} - 7 \sqrt{2} + {\sqrt{2}}^{1} \sqrt{2}}{47}$

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{49 - 7 \sqrt{2} - 7 \sqrt{2} + {\sqrt{2}}^{1} {\sqrt{2}}^{1}}{47}$

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

$\frac{49 - 7 \sqrt{2} - 7 \sqrt{2} + {\sqrt{2}}^{1 + 1}}{47}$

Add $1$ and $1$ .

$\frac{49 - 7 \sqrt{2} - 7 \sqrt{2} + {\sqrt{2}}^{2}}{47}$

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

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

$\frac{49 - 7 \sqrt{2} - 7 \sqrt{2} + {(2^{\frac{1}{2}})}^{2}}{47}$

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

$\frac{49 - 7 \sqrt{2} - 7 \sqrt{2} + 2^{\frac{1}{2} \cdot 2}}{47}$

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

$\frac{49 - 7 \sqrt{2} - 7 \sqrt{2} + 2^{\frac{2}{2}}}{47}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{49 - 7 \sqrt{2} - 7 \sqrt{2} + 2^{\frac{2}{2}}}{47}$

Rewrite the expression.

$\frac{49 - 7 \sqrt{2} - 7 \sqrt{2} + 2^{1}}{47}$

Evaluate the exponent.

$\frac{49 - 7 \sqrt{2} - 7 \sqrt{2} + 2}{47}$

Add $49$ and $2$ .

$\frac{51 - 7 \sqrt{2} - 7 \sqrt{2}}{47}$

Subtract $7 \sqrt{2}$ from $- 7 \sqrt{2}$ .

$\frac{51 - 14 \sqrt{2}}{47}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{51 - 14 \sqrt{2}}{47}$

Decimal Form:

$0.66385127 \dots$

Evaluate (7- square root of 2)/(7+ square root of 2)

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