Simplify (1- square root of 2)/(1+ square root of 2)

$\frac{1 - \sqrt{2}}{1 + \sqrt{2}}$

Step 1

Multiply $\frac{1 - \sqrt{2}}{1 + \sqrt{2}}$ by $\frac{1 - \sqrt{2}}{1 - \sqrt{2}}$ .

$\frac{1 - \sqrt{2}}{1 + \sqrt{2}} \cdot \frac{1 - \sqrt{2}}{1 - \sqrt{2}}$

Step 2

Combine fractions.

Multiply $\frac{1 - \sqrt{2}}{1 + \sqrt{2}}$ by $\frac{1 - \sqrt{2}}{1 - \sqrt{2}}$ .

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

Expand the denominator using the FOIL method.

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

Simplify.

$\frac{(1 - \sqrt{2}) (1 - \sqrt{2})}{- 1}$

Simplify the expression.

Move the negative one from the denominator of $\frac{(1 - \sqrt{2}) (1 - \sqrt{2})}{- 1}$ .

$- 1 \cdot ((1 - \sqrt{2}) (1 - \sqrt{2}))$

Rewrite $- 1 \cdot ((1 - \sqrt{2}) (1 - \sqrt{2}))$ as $- ((1 - \sqrt{2}) (1 - \sqrt{2}))$ .

$- ((1 - \sqrt{2}) (1 - \sqrt{2}))$

Step 3

Expand $(1 - \sqrt{2}) (1 - \sqrt{2})$ using the FOIL Method.

Apply the distributive property.

$- (1 (1 - \sqrt{2}) - \sqrt{2} (1 - \sqrt{2}))$

Apply the distributive property.

$- (1 \cdot 1 + 1 (- \sqrt{2}) - \sqrt{2} (1 - \sqrt{2}))$

Apply the distributive property.

$- (1 \cdot 1 + 1 (- \sqrt{2}) - \sqrt{2} \cdot 1 - \sqrt{2} (- \sqrt{2}))$

Step 4

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

$- (1 + 1 (- \sqrt{2}) - \sqrt{2} \cdot 1 - \sqrt{2} (- \sqrt{2}))$

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

$- (1 - \sqrt{2} - \sqrt{2} \cdot 1 - \sqrt{2} (- \sqrt{2}))$

Multiply $- 1$ by $1$ .

$- (1 - \sqrt{2} - \sqrt{2} - \sqrt{2} (- \sqrt{2}))$

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

Multiply $- 1$ by $- 1$ .

$- (1 - \sqrt{2} - \sqrt{2} + 1 \sqrt{2} \sqrt{2})$

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

$- (1 - \sqrt{2} - \sqrt{2} + \sqrt{2} \sqrt{2})$

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

$- (1 - \sqrt{2} - \sqrt{2} + 2)$

Add $1$ and $2$ .

$- (3 - \sqrt{2} - \sqrt{2})$

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

$- (3 - 2 \sqrt{2})$

Step 5

Apply the distributive property.

$- 1 \cdot 3 - (- 2 \sqrt{2})$

Step 6

Multiply.

Multiply $- 1$ by $3$ .

$- 3 - (- 2 \sqrt{2})$

Multiply $- 2$ by $- 1$ .

$- 3 + 2 \sqrt{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- 3 + 2 \sqrt{2}$

Decimal Form:

$- 0.17157287 \dots$

Simplify (1- square root of 2)/(1+ square root of 2)

Related Questions

Support LCMGCF.com by disabling your adblocker.