Simplify (3+ square root of 8)/(3- square root of 8)

$\frac{3 + \sqrt{8}}{3 - \sqrt{8}}$

Step 1

Simplify the numerator.

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

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

Rewrite $4$ as $2^{2}$ .

$\frac{3 + \sqrt{2^{2} \cdot 2}}{3 - \sqrt{8}}$

Pull terms out from under the radical.

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

Step 2

Simplify the denominator.

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

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

Rewrite $4$ as $2^{2}$ .

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 1$ .

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

Step 3

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

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

Step 4

Combine fractions.

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

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

Expand the denominator using the FOIL method.

$\frac{(3 + 2 \sqrt{2}) (3 + 2 \sqrt{2})}{9 + 6 \sqrt{2} - 6 \sqrt{2} - 4 {\sqrt{2}}^{2}}$

Simplify.

$\frac{(3 + 2 \sqrt{2}) (3 + 2 \sqrt{2})}{1}$

Divide $(3 + 2 \sqrt{2}) (3 + 2 \sqrt{2})$ by $1$ .

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

Step 5

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

Apply the distributive property.

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

Apply the distributive property.

$3 \cdot 3 + 3 (2 \sqrt{2}) + 2 \sqrt{2} (3 + 2 \sqrt{2})$

Apply the distributive property.

$3 \cdot 3 + 3 (2 \sqrt{2}) + 2 \sqrt{2} \cdot 3 + 2 \sqrt{2} (2 \sqrt{2})$

Step 6

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$9 + 3 (2 \sqrt{2}) + 2 \sqrt{2} \cdot 3 + 2 \sqrt{2} (2 \sqrt{2})$

Multiply $2$ by $3$ .

$9 + 6 \sqrt{2} + 2 \sqrt{2} \cdot 3 + 2 \sqrt{2} (2 \sqrt{2})$

Multiply $3$ by $2$ .

$9 + 6 \sqrt{2} + 6 \sqrt{2} + 2 \sqrt{2} (2 \sqrt{2})$

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

Multiply $2$ by $2$ .

$9 + 6 \sqrt{2} + 6 \sqrt{2} + 4 \sqrt{2} \sqrt{2}$

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

$9 + 6 \sqrt{2} + 6 \sqrt{2} + 4 ({\sqrt{2}}^{1} \sqrt{2})$

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

$9 + 6 \sqrt{2} + 6 \sqrt{2} + 4 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})$

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

$9 + 6 \sqrt{2} + 6 \sqrt{2} + 4 {\sqrt{2}}^{1 + 1}$

Add $1$ and $1$ .

$9 + 6 \sqrt{2} + 6 \sqrt{2} + 4 {\sqrt{2}}^{2}$

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

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

$9 + 6 \sqrt{2} + 6 \sqrt{2} + 4 {(2^{\frac{1}{2}})}^{2}$

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

$9 + 6 \sqrt{2} + 6 \sqrt{2} + 4 \cdot 2^{\frac{1}{2} \cdot 2}$

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

$9 + 6 \sqrt{2} + 6 \sqrt{2} + 4 \cdot 2^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$9 + 6 \sqrt{2} + 6 \sqrt{2} + 4 \cdot 2^{\frac{2}{2}}$

Rewrite the expression.

$9 + 6 \sqrt{2} + 6 \sqrt{2} + 4 \cdot 2^{1}$

Evaluate the exponent.

$9 + 6 \sqrt{2} + 6 \sqrt{2} + 4 \cdot 2$

Multiply $4$ by $2$ .

$9 + 6 \sqrt{2} + 6 \sqrt{2} + 8$

Add $9$ and $8$ .

$17 + 6 \sqrt{2} + 6 \sqrt{2}$

Add $6 \sqrt{2}$ and $6 \sqrt{2}$ .

$17 + 12 \sqrt{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$17 + 12 \sqrt{2}$

Decimal Form:

$33.97056274 \dots$

Simplify (3+ square root of 8)/(3- square root of 8)

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