Simplify ( square root of 72+6)/( square root of 2)

$\frac{\sqrt{72} + 6}{\sqrt{2}}$

Step 1

Simplify the numerator.

Rewrite $72$ as $6^{2} \cdot 2$ .

Factor $36$ out of $72$ .

$\frac{\sqrt{36 (2)} + 6}{\sqrt{2}}$

Rewrite $36$ as $6^{2}$ .

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

Pull terms out from under the radical.

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

Step 2

Multiply $\frac{6 \sqrt{2} + 6}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{6 \sqrt{2} + 6}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 3

Combine and simplify the denominator.

Multiply $\frac{6 \sqrt{2} + 6}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 4

Cancel the common factor of $6 \sqrt{2} + 6$ and $2$ .

Factor $2$ out of $(6 \sqrt{2} + 6) \sqrt{2}$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Step 5

Apply the distributive property.

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

Step 6

Multiply $3 \sqrt{2} \sqrt{2}$ .

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

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

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

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

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

$3 {\sqrt{2}}^{1 + 1} + 3 \sqrt{2}$

Add $1$ and $1$ .

$3 {\sqrt{2}}^{2} + 3 \sqrt{2}$

Step 7

Simplify each term.

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$3 \cdot 2^{1} + 3 \sqrt{2}$

Evaluate the exponent.

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

Multiply $3$ by $2$ .

$6 + 3 \sqrt{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$6 + 3 \sqrt{2}$

Decimal Form:

$10.24264068 \dots$

Simplify ( square root of 72+6)/( square root of 2)

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