Basic Math Examples

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

Step 1

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 2

Combine $4$ and $\frac{3}{3}$ .

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Multiply $4$ by $3$ .

$\frac{\sqrt{\frac{12 + 4}{3}}}{2} + \frac{3 - \frac{4}{3}}{4}$

Step 4.2

Add $12$ and $4$ .

$\frac{\sqrt{\frac{16}{3}}}{2} + \frac{3 - \frac{4}{3}}{4}$

Step 5

To write $3$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\sqrt{\frac{16}{3}}}{2} + \frac{3 \cdot \frac{3}{3} - \frac{4}{3}}{4}$

Step 6

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

$\frac{\sqrt{\frac{16}{3}}}{2} + \frac{\frac{3 \cdot 3}{3} - \frac{4}{3}}{4}$

Step 7

Combine the numerators over the common denominator.

$\frac{\sqrt{\frac{16}{3}}}{2} + \frac{\frac{3 \cdot 3 - 4}{3}}{4}$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $3$ by $3$ .

$\frac{\sqrt{\frac{16}{3}}}{2} + \frac{\frac{9 - 4}{3}}{4}$

Step 8.2

Subtract $4$ from $9$ .

$\frac{\sqrt{\frac{16}{3}}}{2} + \frac{\frac{5}{3}}{4}$

Step 9

Simplify each term.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Rewrite $\sqrt{\frac{16}{3}}$ as $\frac{\sqrt{16}}{\sqrt{3}}$ .

$\frac{\frac{\sqrt{16}}{\sqrt{3}}}{2} + \frac{\frac{5}{3}}{4}$

Step 9.1.2

Simplify the numerator.

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Step 9.1.2.1

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

$\frac{\frac{\sqrt{4^{2}}}{\sqrt{3}}}{2} + \frac{\frac{5}{3}}{4}$

Step 9.1.2.2

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

$\frac{\frac{4}{\sqrt{3}}}{2} + \frac{\frac{5}{3}}{4}$

Step 9.1.3

Multiply $\frac{4}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 9.1.4

Combine and simplify the denominator.

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Step 9.1.4.1

Multiply $\frac{4}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\frac{4 \sqrt{3}}{\sqrt{3} \sqrt{3}}}{2} + \frac{\frac{5}{3}}{4}$

Step 9.1.4.2

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

$\frac{\frac{4 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}}{2} + \frac{\frac{5}{3}}{4}$

Step 9.1.4.3

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

$\frac{\frac{4 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}}{2} + \frac{\frac{5}{3}}{4}$

Step 9.1.4.4

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

$\frac{\frac{4 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}}{2} + \frac{\frac{5}{3}}{4}$

Step 9.1.4.5

Add $1$ and $1$ .

$\frac{\frac{4 \sqrt{3}}{{\sqrt{3}}^{2}}}{2} + \frac{\frac{5}{3}}{4}$

Step 9.1.4.6

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

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Step 9.1.4.6.1

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

$\frac{\frac{4 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}}{2} + \frac{\frac{5}{3}}{4}$

Step 9.1.4.6.2

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

$\frac{\frac{4 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}}{2} + \frac{\frac{5}{3}}{4}$

Step 9.1.4.6.3

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

$\frac{\frac{4 \sqrt{3}}{3^{\frac{2}{2}}}}{2} + \frac{\frac{5}{3}}{4}$

Step 9.1.4.6.4

Cancel the common factor of $2$ .

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Step 9.1.4.6.4.1

Cancel the common factor.

$\frac{\frac{4 \sqrt{3}}{3^{\frac{2}{2}}}}{2} + \frac{\frac{5}{3}}{4}$

Step 9.1.4.6.4.2

Rewrite the expression.

$\frac{\frac{4 \sqrt{3}}{3^{1}}}{2} + \frac{\frac{5}{3}}{4}$

Step 9.1.4.6.5

Evaluate the exponent.

$\frac{\frac{4 \sqrt{3}}{3}}{2} + \frac{\frac{5}{3}}{4}$

Step 9.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{4 \sqrt{3}}{3} \cdot \frac{1}{2} + \frac{\frac{5}{3}}{4}$

Step 9.3

Cancel the common factor of $2$ .

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Step 9.3.1

Factor $2$ out of $4 \sqrt{3}$ .

$\frac{2 (2 \sqrt{3})}{3} \cdot \frac{1}{2} + \frac{\frac{5}{3}}{4}$

Step 9.3.2

Cancel the common factor.

$\frac{2 (2 \sqrt{3})}{3} \cdot \frac{1}{2} + \frac{\frac{5}{3}}{4}$

Step 9.3.3

Rewrite the expression.

$\frac{2 \sqrt{3}}{3} + \frac{\frac{5}{3}}{4}$

Step 9.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 \sqrt{3}}{3} + \frac{5}{3} \cdot \frac{1}{4}$

Step 9.5

Multiply $\frac{5}{3} \cdot \frac{1}{4}$ .

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Step 9.5.1

Multiply $\frac{5}{3}$ by $\frac{1}{4}$ .

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

Step 9.5.2

Multiply $3$ by $4$ .

$\frac{2 \sqrt{3}}{3} + \frac{5}{12}$

Step 10

To write $\frac{2 \sqrt{3}}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{2 \sqrt{3}}{3} \cdot \frac{4}{4} + \frac{5}{12}$

Step 11

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.1

Multiply $\frac{2 \sqrt{3}}{3}$ by $\frac{4}{4}$ .

$\frac{2 \sqrt{3} \cdot 4}{3 \cdot 4} + \frac{5}{12}$

Step 11.2

Multiply $3$ by $4$ .

$\frac{2 \sqrt{3} \cdot 4}{12} + \frac{5}{12}$

Step 12

Combine the numerators over the common denominator.

$\frac{2 \sqrt{3} \cdot 4 + 5}{12}$

Step 13

Multiply $4$ by $2$ .

$\frac{8 \sqrt{3} + 5}{12}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{8 \sqrt{3} + 5}{12}$

Decimal Form:

$1.57136720 \dots$