Simplify (i square root of 1 2/3)/(i square root of 2/3)

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

Step 1

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Add $3$ and $2$ .

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

Step 2

Cancel the common factor of $i$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

Simplify the numerator.

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

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $5$ by $3$ .

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

Step 4

Simplify the denominator.

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

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Simplify the numerator.

Combine using the product rule for radicals.

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

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 $3$ .

Cancel the common factor.

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

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 $6$ .

Factor $3$ out of $15$ .

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

Cancel the common factors.

Factor $3$ out of $6$ .

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

Cancel the common factor.

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

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.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 13

Simplify the numerator.

Combine using the product rule for radicals.

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

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:

$1.58113883 \dots$

Simplify (i square root of 1 2/3)/(i square root of 2/3)

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