Algebra Examples

$\frac{1}{\sqrt{2}} + \frac{3}{\sqrt{8}} + \frac{6}{\sqrt{32}}$

Step 1

Find the common denominator.

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Step 1.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{256}}{\sqrt{256}}$ .

$\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{256}}{\sqrt{256}} + \frac{3}{\sqrt{8}} + \frac{6}{\sqrt{32}}$

Step 1.2

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{256}}{\sqrt{256}}$ .

$\frac{\sqrt{256}}{\sqrt{2} \sqrt{256}} + \frac{3}{\sqrt{8}} + \frac{6}{\sqrt{32}}$

Step 1.3

Multiply $\frac{3}{\sqrt{8}}$ by $\frac{\sqrt{64}}{\sqrt{64}}$ .

$\frac{\sqrt{256}}{\sqrt{2} \sqrt{256}} + \frac{3}{\sqrt{8}} \cdot \frac{\sqrt{64}}{\sqrt{64}} + \frac{6}{\sqrt{32}}$

Step 1.4

Multiply $\frac{3}{\sqrt{8}}$ by $\frac{\sqrt{64}}{\sqrt{64}}$ .

$\frac{\sqrt{256}}{\sqrt{2} \sqrt{256}} + \frac{3 \sqrt{64}}{\sqrt{8} \sqrt{64}} + \frac{6}{\sqrt{32}}$

Step 1.5

Multiply $\frac{6}{\sqrt{32}}$ by $\frac{\sqrt{16}}{\sqrt{16}}$ .

$\frac{\sqrt{256}}{\sqrt{2} \sqrt{256}} + \frac{3 \sqrt{64}}{\sqrt{8} \sqrt{64}} + \frac{6}{\sqrt{32}} \cdot \frac{\sqrt{16}}{\sqrt{16}}$

Step 1.6

Multiply $\frac{6}{\sqrt{32}}$ by $\frac{\sqrt{16}}{\sqrt{16}}$ .

$\frac{\sqrt{256}}{\sqrt{2} \sqrt{256}} + \frac{3 \sqrt{64}}{\sqrt{8} \sqrt{64}} + \frac{6 \sqrt{16}}{\sqrt{32} \sqrt{16}}$

Step 1.7

Combine using the product rule for radicals.

$\frac{\sqrt{256}}{\sqrt{2 \cdot 256}} + \frac{3 \sqrt{64}}{\sqrt{8} \sqrt{64}} + \frac{6 \sqrt{16}}{\sqrt{32} \sqrt{16}}$

Step 1.8

Multiply $2$ by $256$ .

$\frac{\sqrt{256}}{\sqrt{512}} + \frac{3 \sqrt{64}}{\sqrt{8} \sqrt{64}} + \frac{6 \sqrt{16}}{\sqrt{32} \sqrt{16}}$

Step 1.9

Combine using the product rule for radicals.

$\frac{\sqrt{256}}{\sqrt{512}} + \frac{3 \sqrt{64}}{\sqrt{8 \cdot 64}} + \frac{6 \sqrt{16}}{\sqrt{32} \sqrt{16}}$

Step 1.10

Multiply $8$ by $64$ .

$\frac{\sqrt{256}}{\sqrt{512}} + \frac{3 \sqrt{64}}{\sqrt{512}} + \frac{6 \sqrt{16}}{\sqrt{32} \sqrt{16}}$

Step 1.11

Reorder the factors of $\sqrt{32} \sqrt{16}$ .

$\frac{\sqrt{256}}{\sqrt{512}} + \frac{3 \sqrt{64}}{\sqrt{512}} + \frac{6 \sqrt{16}}{\sqrt{16} \sqrt{32}}$

Step 1.12

Combine using the product rule for radicals.

$\frac{\sqrt{256}}{\sqrt{512}} + \frac{3 \sqrt{64}}{\sqrt{512}} + \frac{6 \sqrt{16}}{\sqrt{16 \cdot 32}}$

Step 1.13

Multiply $16$ by $32$ .

$\frac{\sqrt{256}}{\sqrt{512}} + \frac{3 \sqrt{64}}{\sqrt{512}} + \frac{6 \sqrt{16}}{\sqrt{512}}$

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{\sqrt{256} + 3 \sqrt{64} + 6 \sqrt{16}}{\sqrt{512}}$

Step 2.2

Simplify each term.

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Step 2.2.1

Rewrite $256$ as $16^{2}$ .

$\frac{\sqrt{16^{2}} + 3 \sqrt{64} + 6 \sqrt{16}}{\sqrt{512}}$

Step 2.2.2

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

$\frac{16 + 3 \sqrt{64} + 6 \sqrt{16}}{\sqrt{512}}$

Step 2.2.3

Rewrite $64$ as $8^{2}$ .

$\frac{16 + 3 \sqrt{8^{2}} + 6 \sqrt{16}}{\sqrt{512}}$

Step 2.2.4

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

$\frac{16 + 3 \cdot 8 + 6 \sqrt{16}}{\sqrt{512}}$

Step 2.2.5

Multiply $3$ by $8$ .

$\frac{16 + 24 + 6 \sqrt{16}}{\sqrt{512}}$

Step 2.2.6

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

$\frac{16 + 24 + 6 \sqrt{4^{2}}}{\sqrt{512}}$

Step 2.2.7

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

$\frac{16 + 24 + 6 \cdot 4}{\sqrt{512}}$

Step 2.2.8

Multiply $6$ by $4$ .

$\frac{16 + 24 + 24}{\sqrt{512}}$

Step 2.3

Simplify by adding numbers.

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Step 2.3.1

Add $16$ and $24$ .

$\frac{40 + 24}{\sqrt{512}}$

Step 2.3.2

Add $40$ and $24$ .

$\frac{64}{\sqrt{512}}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $512$ as $16^{2} \cdot 2$ .

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Step 3.1.1

Factor $256$ out of $512$ .

$\frac{64}{\sqrt{256 (2)}}$

Step 3.1.2

Rewrite $256$ as $16^{2}$ .

$\frac{64}{\sqrt{16^{2} \cdot 2}}$

Step 3.2

Pull terms out from under the radical.

$\frac{64}{16 \sqrt{2}}$

Step 4

Cancel the common factor of $64$ and $16$ .

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Step 4.1

Factor $16$ out of $64$ .

$\frac{16 \cdot 4}{16 \sqrt{2}}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

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

$\frac{16 \cdot 4}{16 (\sqrt{2})}$

Step 4.2.2

Cancel the common factor.

$\frac{16 \cdot 4}{16 \sqrt{2}}$

Step 4.2.3

Rewrite the expression.

$\frac{4}{\sqrt{2}}$

Step 5

Multiply $\frac{4}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{4}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{4}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{4 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 6.2

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

$\frac{4 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 6.3

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

$\frac{4 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 6.4

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

$\frac{4 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 6.5

Add $1$ and $1$ .

$\frac{4 \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 6.6

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

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Step 6.6.1

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

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

Step 6.6.2

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

$\frac{4 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 6.6.3

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

$\frac{4 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 6.6.4

Cancel the common factor of $2$ .

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Step 6.6.4.1

Cancel the common factor.

$\frac{4 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 6.6.4.2

Rewrite the expression.

$\frac{4 \sqrt{2}}{2^{1}}$

Step 6.6.5

Evaluate the exponent.

$\frac{4 \sqrt{2}}{2}$

Step 7

Cancel the common factor of $4$ and $2$ .

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Step 7.1

Factor $2$ out of $4 \sqrt{2}$ .

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

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $2$ out of $2$ .

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

Step 7.2.2

Cancel the common factor.

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

Step 7.2.3

Rewrite the expression.

$\frac{2 \sqrt{2}}{1}$

Step 7.2.4

Divide $2 \sqrt{2}$ by $1$ .

$2 \sqrt{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{2}$

Decimal Form:

$2.82842712 \dots$