Simplify the Radical Expression (4+ square root of 2)/(5 square root of 2-2)

$\frac{4 + \sqrt{2}}{5 \sqrt{2} - 2}$

Step 1

Multiply $\frac{4 + \sqrt{2}}{5 \sqrt{2} - 2}$ by $\frac{5 \sqrt{2} + 2}{5 \sqrt{2} + 2}$ .

$\frac{4 + \sqrt{2}}{5 \sqrt{2} - 2} \cdot \frac{5 \sqrt{2} + 2}{5 \sqrt{2} + 2}$

Step 2

Multiply $\frac{4 + \sqrt{2}}{5 \sqrt{2} - 2}$ by $\frac{5 \sqrt{2} + 2}{5 \sqrt{2} + 2}$ .

$\frac{(4 + \sqrt{2}) (5 \sqrt{2} + 2)}{(5 \sqrt{2} - 2) (5 \sqrt{2} + 2)}$

Step 3

Expand the denominator using the FOIL method.

$\frac{(4 + \sqrt{2}) (5 \sqrt{2} + 2)}{25 {\sqrt{2}}^{2} + 10 \sqrt{2} - 10 \sqrt{2} - 4}$

Step 4

Simplify.

$\frac{(4 + \sqrt{2}) (5 \sqrt{2} + 2)}{46}$

Step 5

Expand $(4 + \sqrt{2}) (5 \sqrt{2} + 2)$ using the FOIL Method.

Apply the distributive property.

$\frac{4 (5 \sqrt{2} + 2) + \sqrt{2} (5 \sqrt{2} + 2)}{46}$

Apply the distributive property.

$\frac{4 (5 \sqrt{2}) + 4 \cdot 2 + \sqrt{2} (5 \sqrt{2} + 2)}{46}$

Apply the distributive property.

$\frac{4 (5 \sqrt{2}) + 4 \cdot 2 + \sqrt{2} (5 \sqrt{2}) + \sqrt{2} \cdot 2}{46}$

Step 6

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $4$ .

$\frac{20 \sqrt{2} + 4 \cdot 2 + \sqrt{2} (5 \sqrt{2}) + \sqrt{2} \cdot 2}{46}$

Multiply $4$ by $2$ .

$\frac{20 \sqrt{2} + 8 + \sqrt{2} (5 \sqrt{2}) + \sqrt{2} \cdot 2}{46}$

Multiply $\sqrt{2} (5 \sqrt{2})$ .

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

$\frac{20 \sqrt{2} + 8 + 5 ({\sqrt{2}}^{1} \sqrt{2}) + \sqrt{2} \cdot 2}{46}$

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

$\frac{20 \sqrt{2} + 8 + 5 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) + \sqrt{2} \cdot 2}{46}$

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

$\frac{20 \sqrt{2} + 8 + 5 {\sqrt{2}}^{1 + 1} + \sqrt{2} \cdot 2}{46}$

Add $1$ and $1$ .

$\frac{20 \sqrt{2} + 8 + 5 {\sqrt{2}}^{2} + \sqrt{2} \cdot 2}{46}$

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

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

$\frac{20 \sqrt{2} + 8 + 5 {(2^{\frac{1}{2}})}^{2} + \sqrt{2} \cdot 2}{46}$

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

$\frac{20 \sqrt{2} + 8 + 5 \cdot 2^{\frac{1}{2} \cdot 2} + \sqrt{2} \cdot 2}{46}$

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

$\frac{20 \sqrt{2} + 8 + 5 \cdot 2^{\frac{2}{2}} + \sqrt{2} \cdot 2}{46}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{20 \sqrt{2} + 8 + 5 \cdot 2^{\frac{2}{2}} + \sqrt{2} \cdot 2}{46}$

Rewrite the expression.

$\frac{20 \sqrt{2} + 8 + 5 \cdot 2^{1} + \sqrt{2} \cdot 2}{46}$

Evaluate the exponent.

$\frac{20 \sqrt{2} + 8 + 5 \cdot 2 + \sqrt{2} \cdot 2}{46}$

Multiply $5$ by $2$ .

$\frac{20 \sqrt{2} + 8 + 10 + \sqrt{2} \cdot 2}{46}$

Move $2$ to the left of $\sqrt{2}$ .

$\frac{20 \sqrt{2} + 8 + 10 + 2 \sqrt{2}}{46}$

Add $20 \sqrt{2}$ and $2 \sqrt{2}$ .

$\frac{8 + 10 + 22 \sqrt{2}}{46}$

Add $8$ and $10$ .

$\frac{18 + 22 \sqrt{2}}{46}$

Step 7

Cancel the common factor of $18 + 22 \sqrt{2}$ and $46$ .

Factor $2$ out of $18$ .

$\frac{2 (9) + 22 \sqrt{2}}{46}$

Factor $2$ out of $22 \sqrt{2}$ .

$\frac{2 (9) + 2 (11 \sqrt{2})}{46}$

Factor $2$ out of $2 (9) + 2 (11 \sqrt{2})$ .

$\frac{2 (9 + 11 \sqrt{2})}{46}$

Cancel the common factors.

Factor $2$ out of $46$ .

$\frac{2 (9 + 11 \sqrt{2})}{2 \cdot 23}$

Cancel the common factor.

$\frac{2 (9 + 11 \sqrt{2})}{2 \cdot 23}$

Rewrite the expression.

$\frac{9 + 11 \sqrt{2}}{23}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{9 + 11 \sqrt{2}}{23}$

Decimal Form:

$1.06766735 \dots$

Simplify the Radical Expression (4+ square root of 2)/(5 square root of 2-2)

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