Algebra Examples

$\frac{i \sqrt{\frac{1}{2}}}{i \sqrt{\frac{3}{2}}}$

Step 1

Cancel the common factor of $i$ .

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Step 1.1

Cancel the common factor.

$\frac{i \sqrt{\frac{1}{2}}}{i \sqrt{\frac{3}{2}}}$

Step 1.2

Rewrite the expression.

$\frac{\sqrt{\frac{1}{2}}}{\sqrt{\frac{3}{2}}}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$\frac{\frac{\sqrt{1}}{\sqrt{2}}}{\sqrt{\frac{3}{2}}}$

Step 2.2

Any root of $1$ is $1$ .

$\frac{\frac{1}{\sqrt{2}}}{\sqrt{\frac{3}{2}}}$

Step 2.3

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

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

Step 2.4

Combine and simplify the denominator.

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Step 2.4.1

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

$\frac{\frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}}{\sqrt{\frac{3}{2}}}$

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.4.4

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

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

Step 2.4.5

Add $1$ and $1$ .

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

Step 2.4.6

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

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Step 2.4.6.1

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

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

Step 2.4.6.2

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

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

Step 2.4.6.3

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

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

Step 2.4.6.4

Cancel the common factor of $2$ .

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Step 2.4.6.4.1

Cancel the common factor.

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

Step 2.4.6.4.2

Rewrite the expression.

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

Step 2.4.6.5

Evaluate the exponent.

$\frac{\frac{\sqrt{2}}{2}}{\sqrt{\frac{3}{2}}}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $\sqrt{\frac{3}{2}}$ as $\frac{\sqrt{3}}{\sqrt{2}}$ .

$\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{3}}{\sqrt{2}}}$

Step 3.2

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

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

Step 3.3

Combine and simplify the denominator.

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Step 3.3.1

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

$\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{3} \sqrt{2}}{\sqrt{2} \sqrt{2}}}$

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

Add $1$ and $1$ .

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

Step 3.3.6

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

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Step 3.3.6.1

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

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

Step 3.3.6.2

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

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

Step 3.3.6.3

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

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

Step 3.3.6.4

Cancel the common factor of $2$ .

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Step 3.3.6.4.1

Cancel the common factor.

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

Step 3.3.6.4.2

Rewrite the expression.

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

Step 3.3.6.5

Evaluate the exponent.

$\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{3} \sqrt{2}}{2}}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Combine using the product rule for radicals.

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

Step 3.4.2

Multiply $3$ by $2$ .

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Cancel the common factor of $2$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

$\sqrt{2} \frac{1}{\sqrt{6}}$

Step 6

Combine $\sqrt{2}$ and $\frac{1}{\sqrt{6}}$ .

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

Step 7

Combine $\sqrt{2}$ and $\sqrt{6}$ into a single radical.

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

Step 8

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

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Step 8.1

Factor $2$ out of $2$ .

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

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $2$ out of $6$ .

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

Step 8.2.2

Cancel the common factor.

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

Step 8.2.3

Rewrite the expression.

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

Step 9

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

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

Step 10

Any root of $1$ is $1$ .

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

Step 11

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 12

Combine and simplify the denominator.

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Step 12.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 12.2

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 12.3

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 12.4

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

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

Step 12.5

Add $1$ and $1$ .

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

Step 12.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 12.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 12.6.2

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

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

Step 12.6.3

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

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

Step 12.6.4

Cancel the common factor of $2$ .

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Step 12.6.4.1

Cancel the common factor.

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

Step 12.6.4.2

Rewrite the expression.

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

Step 12.6.5

Evaluate the exponent.

$\frac{\sqrt{3}}{3}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{3}}{3}$

Decimal Form:

$0.57735026 \dots$