Basic Math Examples

$\sqrt{(\frac{2}{3} - \frac{4}{3}) (\frac{5}{4} - \frac{3}{4})}$

Step 1

Simplify the expression.

Tap for more steps...

Step 1.1

Combine the numerators over the common denominator.

$\sqrt{\frac{2 - 4}{3} (\frac{5}{4} - \frac{3}{4})}$

Step 1.2

Subtract $4$ from $2$ .

$\sqrt{\frac{- 2}{3} (\frac{5}{4} - \frac{3}{4})}$

Step 1.3

Move the negative in front of the fraction.

$\sqrt{- \frac{2}{3} (\frac{5}{4} - \frac{3}{4})}$

Step 1.4

Combine the numerators over the common denominator.

$\sqrt{- \frac{2}{3} \cdot \frac{5 - 3}{4}}$

Step 1.5

Subtract $3$ from $5$ .

$\sqrt{- \frac{2}{3} \cdot \frac{2}{4}}$

Step 2

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

Tap for more steps...

Step 2.1

Factor $2$ out of $2$ .

$\sqrt{- \frac{2}{3} \cdot \frac{2 (1)}{4}}$

Step 2.2

Cancel the common factors.

Tap for more steps...

Step 2.2.1

Factor $2$ out of $4$ .

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

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

Step 3

Combine exponents.

Tap for more steps...

Step 3.1

Multiply $\frac{1}{2}$ by $\frac{2}{3}$ .

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

Step 3.2

Multiply $2$ by $3$ .

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

Step 4

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

Tap for more steps...

Step 4.1

Factor $2$ out of $2$ .

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

Step 4.2

Cancel the common factors.

Tap for more steps...

Step 4.2.1

Factor $2$ out of $6$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 5

Rewrite $- \frac{1}{3}$ as $i^{2} \frac{1^{2}}{3}$ .

Tap for more steps...

Step 5.1

Rewrite $- 1$ as $i^{2}$ .

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

Step 5.2

Rewrite $1$ as $1^{2}$ .

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

Step 6

Pull terms out from under the radical.

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

Step 7

One to any power is one.

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

Step 8

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

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

Step 9

Any root of $1$ is $1$ .

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

Step 10

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

$i (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 11

Combine and simplify the denominator.

Tap for more steps...

Step 11.1

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

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

Step 11.2

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

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

Step 11.3

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

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

Step 11.4

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

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

Step 11.5

Add $1$ and $1$ .

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

Step 11.6

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

Tap for more steps...

Step 11.6.1

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

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

Step 11.6.2

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

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

Step 11.6.3

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

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

Step 11.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 11.6.4.1

Cancel the common factor.

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

Step 11.6.4.2

Rewrite the expression.

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

Step 11.6.5

Evaluate the exponent.

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

Step 12

Combine $i$ and $\frac{\sqrt{3}}{3}$ .

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