Algebra Examples

$(8 \sqrt{12} + 3 \sqrt{18} - \sqrt{32}) \div \sqrt{2}$

Step 1

Rewrite the division as a fraction.

$\frac{8 \sqrt{12} + 3 \sqrt{18} - \sqrt{32}}{\sqrt{2}}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.1.1

Factor $4$ out of $12$ .

$\frac{8 \sqrt{4 (3)} + 3 \sqrt{18} - \sqrt{32}}{\sqrt{2}}$

Step 2.1.2

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

$\frac{8 \sqrt{2^{2} \cdot 3} + 3 \sqrt{18} - \sqrt{32}}{\sqrt{2}}$

Step 2.2

Pull terms out from under the radical.

$\frac{8 (2 \sqrt{3}) + 3 \sqrt{18} - \sqrt{32}}{\sqrt{2}}$

Step 2.3

Multiply $2$ by $8$ .

$\frac{16 \sqrt{3} + 3 \sqrt{18} - \sqrt{32}}{\sqrt{2}}$

Step 2.4

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 2.4.1

Factor $9$ out of $18$ .

$\frac{16 \sqrt{3} + 3 \sqrt{9 (2)} - \sqrt{32}}{\sqrt{2}}$

Step 2.4.2

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

$\frac{16 \sqrt{3} + 3 \sqrt{3^{2} \cdot 2} - \sqrt{32}}{\sqrt{2}}$

Step 2.5

Pull terms out from under the radical.

$\frac{16 \sqrt{3} + 3 (3 \sqrt{2}) - \sqrt{32}}{\sqrt{2}}$

Step 2.6

Multiply $3$ by $3$ .

$\frac{16 \sqrt{3} + 9 \sqrt{2} - \sqrt{32}}{\sqrt{2}}$

Step 2.7

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 2.7.1

Factor $16$ out of $32$ .

$\frac{16 \sqrt{3} + 9 \sqrt{2} - \sqrt{16 (2)}}{\sqrt{2}}$

Step 2.7.2

Rewrite $16$ as $4^{2}$ .

$\frac{16 \sqrt{3} + 9 \sqrt{2} - \sqrt{4^{2} \cdot 2}}{\sqrt{2}}$

Step 2.8

Pull terms out from under the radical.

$\frac{16 \sqrt{3} + 9 \sqrt{2} - (4 \sqrt{2})}{\sqrt{2}}$

Step 2.9

Multiply $4$ by $- 1$ .

$\frac{16 \sqrt{3} + 9 \sqrt{2} - 4 \sqrt{2}}{\sqrt{2}}$

Step 2.10

Subtract $4 \sqrt{2}$ from $9 \sqrt{2}$ .

$\frac{16 \sqrt{3} + 5 \sqrt{2}}{\sqrt{2}}$

Step 3

Multiply $\frac{16 \sqrt{3} + 5 \sqrt{2}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{16 \sqrt{3} + 5 \sqrt{2}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{16 \sqrt{3} + 5 \sqrt{2}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{(16 \sqrt{3} + 5 \sqrt{2}) \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 4.2

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

$\frac{(16 \sqrt{3} + 5 \sqrt{2}) \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 4.3

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

$\frac{(16 \sqrt{3} + 5 \sqrt{2}) \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 4.4

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

$\frac{(16 \sqrt{3} + 5 \sqrt{2}) \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 4.5

Add $1$ and $1$ .

$\frac{(16 \sqrt{3} + 5 \sqrt{2}) \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 4.6

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

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Step 4.6.1

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

$\frac{(16 \sqrt{3} + 5 \sqrt{2}) \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 4.6.2

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

$\frac{(16 \sqrt{3} + 5 \sqrt{2}) \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 4.6.3

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

$\frac{(16 \sqrt{3} + 5 \sqrt{2}) \sqrt{2}}{2^{\frac{2}{2}}}$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$\frac{(16 \sqrt{3} + 5 \sqrt{2}) \sqrt{2}}{2^{\frac{2}{2}}}$

Step 4.6.4.2

Rewrite the expression.

$\frac{(16 \sqrt{3} + 5 \sqrt{2}) \sqrt{2}}{2^{1}}$

Step 4.6.5

Evaluate the exponent.

$\frac{(16 \sqrt{3} + 5 \sqrt{2}) \sqrt{2}}{2}$

Step 5

Apply the distributive property.

$\frac{16 \sqrt{3} \sqrt{2} + 5 \sqrt{2} \sqrt{2}}{2}$

Step 6

Multiply $16 \sqrt{3} \sqrt{2}$ .

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Step 6.1

Combine using the product rule for radicals.

$\frac{16 \sqrt{2 \cdot 3} + 5 \sqrt{2} \sqrt{2}}{2}$

Step 6.2

Multiply $2$ by $3$ .

$\frac{16 \sqrt{6} + 5 \sqrt{2} \sqrt{2}}{2}$

Step 7

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

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Step 7.1

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

$\frac{16 \sqrt{6} + 5 ({\sqrt{2}}^{1} \sqrt{2})}{2}$

Step 7.2

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

$\frac{16 \sqrt{6} + 5 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{2}$

Step 7.3

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

$\frac{16 \sqrt{6} + 5 {\sqrt{2}}^{1 + 1}}{2}$

Step 7.4

Add $1$ and $1$ .

$\frac{16 \sqrt{6} + 5 {\sqrt{2}}^{2}}{2}$

Step 8

Simplify each term.

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Step 8.1

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

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Step 8.1.1

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

$\frac{16 \sqrt{6} + 5 {(2^{\frac{1}{2}})}^{2}}{2}$

Step 8.1.2

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

$\frac{16 \sqrt{6} + 5 \cdot 2^{\frac{1}{2} \cdot 2}}{2}$

Step 8.1.3

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

$\frac{16 \sqrt{6} + 5 \cdot 2^{\frac{2}{2}}}{2}$

Step 8.1.4

Cancel the common factor of $2$ .

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Step 8.1.4.1

Cancel the common factor.

$\frac{16 \sqrt{6} + 5 \cdot 2^{\frac{2}{2}}}{2}$

Step 8.1.4.2

Rewrite the expression.

$\frac{16 \sqrt{6} + 5 \cdot 2^{1}}{2}$

Step 8.1.5

Evaluate the exponent.

$\frac{16 \sqrt{6} + 5 \cdot 2}{2}$

Step 8.2

Multiply $5$ by $2$ .

$\frac{16 \sqrt{6} + 10}{2}$

Step 9

Cancel the common factor of $16 \sqrt{6} + 10$ and $2$ .

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Step 9.1

Factor $2$ out of $16 \sqrt{6}$ .

$\frac{2 (8 \sqrt{6}) + 10}{2}$

Step 9.2

Factor $2$ out of $10$ .

$\frac{2 (8 \sqrt{6}) + 2 \cdot 5}{2}$

Step 9.3

Factor $2$ out of $2 (8 \sqrt{6}) + 2 (5)$ .

$\frac{2 (8 \sqrt{6} + 5)}{2}$

Step 9.4

Cancel the common factors.

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Step 9.4.1

Factor $2$ out of $2$ .

$\frac{2 (8 \sqrt{6} + 5)}{2 (1)}$

Step 9.4.2

Cancel the common factor.

$\frac{2 (8 \sqrt{6} + 5)}{2 \cdot 1}$

Step 9.4.3

Rewrite the expression.

$\frac{8 \sqrt{6} + 5}{1}$

Step 9.4.4

Divide $8 \sqrt{6} + 5$ by $1$ .

$8 \sqrt{6} + 5$

Step 10

The result can be shown in multiple forms.

Exact Form:

$8 \sqrt{6} + 5$

Decimal Form:

$24.59591794 \dots$