Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

Rewrite $\sqrt{\frac{9}{2}}$ as $\frac{\sqrt{9}}{\sqrt{2}}$ .

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

Step 1.2

Simplify the numerator.

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Step 1.2.1

Rewrite $9$ as $3^{2}$ .

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

Step 1.2.2

Pull terms out from under the radical, assuming positive real numbers.

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

Step 1.3

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

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

Step 1.4

Combine and simplify the denominator.

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Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.4.3

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

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

Step 1.4.4

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

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

Step 1.4.5

Add $1$ and $1$ .

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

Step 1.4.6

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

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Step 1.4.6.1

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

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

Step 1.4.6.2

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

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

Step 1.4.6.3

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

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

Step 1.4.6.4

Cancel the common factor of $2$ .

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Step 1.4.6.4.1

Cancel the common factor.

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

Step 1.4.6.4.2

Rewrite the expression.

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

Step 1.4.6.5

Evaluate the exponent.

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

Step 1.5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.5.1

Apply the product rule to $\frac{3 \sqrt{2}}{2}$ .

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

Step 1.5.2

Apply the product rule to $3 \sqrt{2}$ .

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

Step 1.6

Simplify the numerator.

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Step 1.6.1

Raise $3$ to the power of $2$ .

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

Step 1.6.2

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

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Step 1.6.2.1

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

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

Step 1.6.2.2

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

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

Step 1.6.2.3

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

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

Step 1.6.2.4

Cancel the common factor of $2$ .

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Step 1.6.2.4.1

Cancel the common factor.

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

Step 1.6.2.4.2

Rewrite the expression.

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

Step 1.6.2.5

Evaluate the exponent.

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

Step 1.7

Raise $2$ to the power of $2$ .

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

Step 1.8

Multiply $9$ by $2$ .

$\frac{18}{4} + 6^{2}$

Step 1.9

Cancel the common factor of $18$ and $4$ .

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Step 1.9.1

Factor $2$ out of $18$ .

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

Step 1.9.2

Cancel the common factors.

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Step 1.9.2.1

Factor $2$ out of $4$ .

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

Step 1.9.2.2

Cancel the common factor.

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

Step 1.9.2.3

Rewrite the expression.

$\frac{9}{2} + 6^{2}$

Step 1.10

Raise $6$ to the power of $2$ .

$\frac{9}{2} + 36$

Step 2

To write $36$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{9}{2} + 36 \cdot \frac{2}{2}$

Step 3

Combine $36$ and $\frac{2}{2}$ .

$\frac{9}{2} + \frac{36 \cdot 2}{2}$

Step 4

Combine the numerators over the common denominator.

$\frac{9 + 36 \cdot 2}{2}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $36$ by $2$ .

$\frac{9 + 72}{2}$

Step 5.2

Add $9$ and $72$ .

$\frac{81}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{81}{2}$

Decimal Form:

$40.5$

Mixed Number Form:

$40 \frac{1}{2}$