Algebra Examples

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

Step 1

Convert $1 \frac{2}{3}$ to an improper fraction.

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Step 1.1

A mixed number is an addition of its whole and fractional parts.

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

Step 1.2

Add $1$ and $\frac{2}{3}$ .

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Step 1.2.1

Write $1$ as a fraction with a common denominator.

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

Step 1.2.2

Combine the numerators over the common denominator.

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

Step 1.2.3

Add $3$ and $2$ .

$\frac{i \sqrt{\frac{5}{3}}}{i \sqrt{\frac{2}{3}}}$

Step 2

Cancel the common factor of $i$ .

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Step 2.1

Cancel the common factor.

$\frac{i \sqrt{\frac{5}{3}}}{i \sqrt{\frac{2}{3}}}$

Step 2.2

Rewrite the expression.

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

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $\sqrt{\frac{5}{3}}$ as $\frac{\sqrt{5}}{\sqrt{3}}$ .

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

Step 3.2

Multiply $\frac{\sqrt{5}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 3.3

Combine and simplify the denominator.

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Step 3.3.1

Multiply $\frac{\sqrt{5}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 3.3.2

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

$\frac{\frac{\sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}}{\sqrt{\frac{2}{3}}}$

Step 3.3.3

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

$\frac{\frac{\sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}}{\sqrt{\frac{2}{3}}}$

Step 3.3.4

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

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

Step 3.3.5

Add $1$ and $1$ .

$\frac{\frac{\sqrt{5} \sqrt{3}}{{\sqrt{3}}^{2}}}{\sqrt{\frac{2}{3}}}$

Step 3.3.6

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

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Step 3.3.6.1

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

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

Step 3.3.6.2

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

$\frac{\frac{\sqrt{5} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}}{\sqrt{\frac{2}{3}}}$

Step 3.3.6.3

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

$\frac{\frac{\sqrt{5} \sqrt{3}}{3^{\frac{2}{2}}}}{\sqrt{\frac{2}{3}}}$

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.

$\frac{\frac{\sqrt{5} \sqrt{3}}{3^{\frac{2}{2}}}}{\sqrt{\frac{2}{3}}}$

Step 3.3.6.4.2

Rewrite the expression.

$\frac{\frac{\sqrt{5} \sqrt{3}}{3^{1}}}{\sqrt{\frac{2}{3}}}$

Step 3.3.6.5

Evaluate the exponent.

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

Step 3.4

Simplify the numerator.

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Step 3.4.1

Combine using the product rule for radicals.

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

Step 3.4.2

Multiply $5$ by $3$ .

$\frac{\frac{\sqrt{15}}{3}}{\sqrt{\frac{2}{3}}}$

Step 4

Simplify the denominator.

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Step 4.1

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

$\frac{\frac{\sqrt{15}}{3}}{\frac{\sqrt{2}}{\sqrt{3}}}$

Step 4.2

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

$\frac{\frac{\sqrt{15}}{3}}{\frac{\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}}$

Step 4.3

Combine and simplify the denominator.

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Step 4.3.1

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

$\frac{\frac{\sqrt{15}}{3}}{\frac{\sqrt{2} \sqrt{3}}{\sqrt{3} \sqrt{3}}}$

Step 4.3.2

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

$\frac{\frac{\sqrt{15}}{3}}{\frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}}$

Step 4.3.3

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

$\frac{\frac{\sqrt{15}}{3}}{\frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}}$

Step 4.3.4

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

$\frac{\frac{\sqrt{15}}{3}}{\frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}}$

Step 4.3.5

Add $1$ and $1$ .

$\frac{\frac{\sqrt{15}}{3}}{\frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{2}}}$

Step 4.3.6

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

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Step 4.3.6.1

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

$\frac{\frac{\sqrt{15}}{3}}{\frac{\sqrt{2} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}}$

Step 4.3.6.2

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

$\frac{\frac{\sqrt{15}}{3}}{\frac{\sqrt{2} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}}$

Step 4.3.6.3

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

$\frac{\frac{\sqrt{15}}{3}}{\frac{\sqrt{2} \sqrt{3}}{3^{\frac{2}{2}}}}$

Step 4.3.6.4

Cancel the common factor of $2$ .

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Step 4.3.6.4.1

Cancel the common factor.

$\frac{\frac{\sqrt{15}}{3}}{\frac{\sqrt{2} \sqrt{3}}{3^{\frac{2}{2}}}}$

Step 4.3.6.4.2

Rewrite the expression.

$\frac{\frac{\sqrt{15}}{3}}{\frac{\sqrt{2} \sqrt{3}}{3^{1}}}$

Step 4.3.6.5

Evaluate the exponent.

$\frac{\frac{\sqrt{15}}{3}}{\frac{\sqrt{2} \sqrt{3}}{3}}$

Step 4.4

Simplify the numerator.

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Step 4.4.1

Combine using the product rule for radicals.

$\frac{\frac{\sqrt{15}}{3}}{\frac{\sqrt{2 \cdot 3}}{3}}$

Step 4.4.2

Multiply $2$ by $3$ .

$\frac{\frac{\sqrt{15}}{3}}{\frac{\sqrt{6}}{3}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{15}}{3} \cdot \frac{3}{\sqrt{6}}$

Step 6

Cancel the common factor of $3$ .

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Step 6.1

Cancel the common factor.

$\frac{\sqrt{15}}{3} \cdot \frac{3}{\sqrt{6}}$

Step 6.2

Rewrite the expression.

$\sqrt{15} \frac{1}{\sqrt{6}}$

Step 7

Combine $\sqrt{15}$ and $\frac{1}{\sqrt{6}}$ .

$\frac{\sqrt{15}}{\sqrt{6}}$

Step 8

Combine $\sqrt{15}$ and $\sqrt{6}$ into a single radical.

$\sqrt{\frac{15}{6}}$

Step 9

Cancel the common factor of $15$ and $6$ .

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Step 9.1

Factor $3$ out of $15$ .

$\sqrt{\frac{3 (5)}{6}}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $3$ out of $6$ .

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

Step 9.2.2

Cancel the common factor.

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

Step 9.2.3

Rewrite the expression.

$\sqrt{\frac{5}{2}}$

Step 10

Rewrite $\sqrt{\frac{5}{2}}$ as $\frac{\sqrt{5}}{\sqrt{2}}$ .

$\frac{\sqrt{5}}{\sqrt{2}}$

Step 11

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

$\frac{\sqrt{5}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 12

Combine and simplify the denominator.

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Step 12.1

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

$\frac{\sqrt{5} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 12.2

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

$\frac{\sqrt{5} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 12.3

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

$\frac{\sqrt{5} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 12.4

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

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

Step 12.5

Add $1$ and $1$ .

$\frac{\sqrt{5} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 12.6

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

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Step 12.6.1

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

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

Step 12.6.2

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

$\frac{\sqrt{5} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 12.6.3

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

$\frac{\sqrt{5} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 12.6.4

Cancel the common factor of $2$ .

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Step 12.6.4.1

Cancel the common factor.

$\frac{\sqrt{5} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 12.6.4.2

Rewrite the expression.

$\frac{\sqrt{5} \sqrt{2}}{2^{1}}$

Step 12.6.5

Evaluate the exponent.

$\frac{\sqrt{5} \sqrt{2}}{2}$

Step 13

Simplify the numerator.

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Step 13.1

Combine using the product rule for radicals.

$\frac{\sqrt{5 \cdot 2}}{2}$

Step 13.2

Multiply $5$ by $2$ .

$\frac{\sqrt{10}}{2}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{10}}{2}$

Decimal Form:

$1.58113883 \dots$