Simplify (3- square root of 2)/(4-2 square root of 2)

$\frac{3 - \sqrt{2}}{4 - 2 \sqrt{2}}$

Step 1

Multiply $\frac{3 - \sqrt{2}}{4 - 2 \sqrt{2}}$ by $\frac{4 + 2 \sqrt{2}}{4 + 2 \sqrt{2}}$ .

$\frac{3 - \sqrt{2}}{4 - 2 \sqrt{2}} \cdot \frac{4 + 2 \sqrt{2}}{4 + 2 \sqrt{2}}$

Step 2

Simplify terms.

Multiply $\frac{3 - \sqrt{2}}{4 - 2 \sqrt{2}}$ by $\frac{4 + 2 \sqrt{2}}{4 + 2 \sqrt{2}}$ .

$\frac{(3 - \sqrt{2}) (4 + 2 \sqrt{2})}{(4 - 2 \sqrt{2}) (4 + 2 \sqrt{2})}$

Expand the denominator using the FOIL method.

$\frac{(3 - \sqrt{2}) (4 + 2 \sqrt{2})}{16 + 8 \sqrt{2} - 8 \sqrt{2} - 4 {\sqrt{2}}^{2}}$

Simplify.

$\frac{(3 - \sqrt{2}) (4 + 2 \sqrt{2})}{8}$

Cancel the common factor of $4 + 2 \sqrt{2}$ and $8$ .

Factor $2$ out of $(3 - \sqrt{2}) (4 + 2 \sqrt{2})$ .

$\frac{2 ((3 - \sqrt{2}) (2 + \sqrt{2}))}{8}$

Cancel the common factors.

Factor $2$ out of $8$ .

$\frac{2 ((3 - \sqrt{2}) (2 + \sqrt{2}))}{2 (4)}$

Cancel the common factor.

$\frac{2 ((3 - \sqrt{2}) (2 + \sqrt{2}))}{2 \cdot 4}$

Rewrite the expression.

$\frac{(3 - \sqrt{2}) (2 + \sqrt{2})}{4}$

Step 3

Expand $(3 - \sqrt{2}) (2 + \sqrt{2})$ using the FOIL Method.

Apply the distributive property.

$\frac{3 (2 + \sqrt{2}) - \sqrt{2} (2 + \sqrt{2})}{4}$

Apply the distributive property.

$\frac{3 \cdot 2 + 3 \sqrt{2} - \sqrt{2} (2 + \sqrt{2})}{4}$

Apply the distributive property.

$\frac{3 \cdot 2 + 3 \sqrt{2} - \sqrt{2} \cdot 2 - \sqrt{2} \sqrt{2}}{4}$

Step 4

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $2$ .

$\frac{6 + 3 \sqrt{2} - \sqrt{2} \cdot 2 - \sqrt{2} \sqrt{2}}{4}$

Multiply $2$ by $- 1$ .

$\frac{6 + 3 \sqrt{2} - 2 \sqrt{2} - \sqrt{2} \sqrt{2}}{4}$

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

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

$\frac{6 + 3 \sqrt{2} - 2 \sqrt{2} - ({\sqrt{2}}^{1} \sqrt{2})}{4}$

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

$\frac{6 + 3 \sqrt{2} - 2 \sqrt{2} - ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{4}$

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

$\frac{6 + 3 \sqrt{2} - 2 \sqrt{2} - {\sqrt{2}}^{1 + 1}}{4}$

Add $1$ and $1$ .

$\frac{6 + 3 \sqrt{2} - 2 \sqrt{2} - {\sqrt{2}}^{2}}{4}$

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

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

$\frac{6 + 3 \sqrt{2} - 2 \sqrt{2} - {(2^{\frac{1}{2}})}^{2}}{4}$

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

$\frac{6 + 3 \sqrt{2} - 2 \sqrt{2} - 2^{\frac{1}{2} \cdot 2}}{4}$

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

$\frac{6 + 3 \sqrt{2} - 2 \sqrt{2} - 2^{\frac{2}{2}}}{4}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{6 + 3 \sqrt{2} - 2 \sqrt{2} - 2^{\frac{2}{2}}}{4}$

Rewrite the expression.

$\frac{6 + 3 \sqrt{2} - 2 \sqrt{2} - 2^{1}}{4}$

Evaluate the exponent.

$\frac{6 + 3 \sqrt{2} - 2 \sqrt{2} - 1 \cdot 2}{4}$

Multiply $- 1$ by $2$ .

$\frac{6 + 3 \sqrt{2} - 2 \sqrt{2} - 2}{4}$

Subtract $2$ from $6$ .

$\frac{4 + 3 \sqrt{2} - 2 \sqrt{2}}{4}$

Subtract $2 \sqrt{2}$ from $3 \sqrt{2}$ .

$\frac{4 + \sqrt{2}}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{4 + \sqrt{2}}{4}$

Decimal Form:

$1.35355339 \dots$

Simplify (3- square root of 2)/(4-2 square root of 2)

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