Basic Math Examples

$\frac{1}{2} \cdot \sqrt{486} - \sqrt{\frac{27}{2}}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $486$ as $9^{2} \cdot 6$ .

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Step 1.1.1

Factor $81$ out of $486$ .

$\frac{1}{2} \cdot \sqrt{81 (6)} - \sqrt{\frac{27}{2}}$

Step 1.1.2

Rewrite $81$ as $9^{2}$ .

$\frac{1}{2} \cdot \sqrt{9^{2} \cdot 6} - \sqrt{\frac{27}{2}}$

Step 1.2

Pull terms out from under the radical.

$\frac{1}{2} \cdot (9 \sqrt{6}) - \sqrt{\frac{27}{2}}$

Step 1.3

Multiply $\frac{1}{2} (9 \sqrt{6})$ .

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Step 1.3.1

Combine $9$ and $\frac{1}{2}$ .

$\frac{9}{2} \sqrt{6} - \sqrt{\frac{27}{2}}$

Step 1.3.2

Combine $\frac{9}{2}$ and $\sqrt{6}$ .

$\frac{9 \sqrt{6}}{2} - \sqrt{\frac{27}{2}}$

Step 1.4

Rewrite $\sqrt{\frac{27}{2}}$ as $\frac{\sqrt{27}}{\sqrt{2}}$ .

$\frac{9 \sqrt{6}}{2} - \frac{\sqrt{27}}{\sqrt{2}}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 1.5.1.1

Factor $9$ out of $27$ .

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

Step 1.5.1.2

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

$\frac{9 \sqrt{6}}{2} - \frac{\sqrt{3^{2} \cdot 3}}{\sqrt{2}}$

Step 1.5.2

Pull terms out from under the radical.

$\frac{9 \sqrt{6}}{2} - \frac{3 \sqrt{3}}{\sqrt{2}}$

Step 1.6

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

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

Step 1.7

Combine and simplify the denominator.

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Step 1.7.1

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

$\frac{9 \sqrt{6}}{2} - \frac{3 \sqrt{3} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 1.7.2

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

$\frac{9 \sqrt{6}}{2} - \frac{3 \sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 1.7.3

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

$\frac{9 \sqrt{6}}{2} - \frac{3 \sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 1.7.4

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

$\frac{9 \sqrt{6}}{2} - \frac{3 \sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 1.7.5

Add $1$ and $1$ .

$\frac{9 \sqrt{6}}{2} - \frac{3 \sqrt{3} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 1.7.6

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

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Step 1.7.6.1

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

$\frac{9 \sqrt{6}}{2} - \frac{3 \sqrt{3} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 1.7.6.2

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

$\frac{9 \sqrt{6}}{2} - \frac{3 \sqrt{3} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 1.7.6.3

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

$\frac{9 \sqrt{6}}{2} - \frac{3 \sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.7.6.4

Cancel the common factor of $2$ .

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Step 1.7.6.4.1

Cancel the common factor.

$\frac{9 \sqrt{6}}{2} - \frac{3 \sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.7.6.4.2

Rewrite the expression.

$\frac{9 \sqrt{6}}{2} - \frac{3 \sqrt{3} \sqrt{2}}{2^{1}}$

Step 1.7.6.5

Evaluate the exponent.

$\frac{9 \sqrt{6}}{2} - \frac{3 \sqrt{3} \sqrt{2}}{2}$

Step 1.8

Simplify the numerator.

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Step 1.8.1

Combine using the product rule for radicals.

$\frac{9 \sqrt{6}}{2} - \frac{3 \sqrt{2 \cdot 3}}{2}$

Step 1.8.2

Multiply $2$ by $3$ .

$\frac{9 \sqrt{6}}{2} - \frac{3 \sqrt{6}}{2}$

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{9 \sqrt{6} - 3 \sqrt{6}}{2}$

Step 2.2

Subtract $3 \sqrt{6}$ from $9 \sqrt{6}$ .

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

Step 2.3

Cancel the common factor of $6$ and $2$ .

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Step 2.3.1

Factor $2$ out of $6 \sqrt{6}$ .

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

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $2$ out of $2$ .

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

Step 2.3.2.2

Cancel the common factor.

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

Step 2.3.2.3

Rewrite the expression.

$\frac{3 \sqrt{6}}{1}$

Step 2.3.2.4

Divide $3 \sqrt{6}$ by $1$ .

$3 \sqrt{6}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$3 \sqrt{6}$

Decimal Form:

$7.34846922 \dots$