Basic Math Examples

$\frac{14}{125} + \sqrt{\frac{5}{3}} - \frac{11}{25}$

Step 1

Find the common denominator.

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Step 1.1

Write $\sqrt{\frac{5}{3}}$ as a fraction with denominator $1$ .

$\frac{14}{125} + \frac{\sqrt{\frac{5}{3}}}{1} - \frac{11}{25}$

Step 1.2

Multiply $\frac{\sqrt{\frac{5}{3}}}{1}$ by $\frac{125}{125}$ .

$\frac{14}{125} + \frac{\sqrt{\frac{5}{3}}}{1} \cdot \frac{125}{125} - \frac{11}{25}$

Step 1.3

Multiply $\frac{\sqrt{\frac{5}{3}}}{1}$ by $\frac{125}{125}$ .

$\frac{14}{125} + \frac{\sqrt{\frac{5}{3}} \cdot 125}{125} - \frac{11}{25}$

Step 1.4

Multiply $\frac{11}{25}$ by $\frac{5}{5}$ .

$\frac{14}{125} + \frac{\sqrt{\frac{5}{3}} \cdot 125}{125} - (\frac{11}{25} \cdot \frac{5}{5})$

Step 1.5

Multiply $\frac{11}{25}$ by $\frac{5}{5}$ .

$\frac{14}{125} + \frac{\sqrt{\frac{5}{3}} \cdot 125}{125} - \frac{11 \cdot 5}{25 \cdot 5}$

Step 1.6

Reorder the factors of $25 \cdot 5$ .

$\frac{14}{125} + \frac{\sqrt{\frac{5}{3}} \cdot 125}{125} - \frac{11 \cdot 5}{5 \cdot 25}$

Step 1.7

Multiply $5$ by $25$ .

$\frac{14}{125} + \frac{\sqrt{\frac{5}{3}} \cdot 125}{125} - \frac{11 \cdot 5}{125}$

Step 2

Combine the numerators over the common denominator.

$\frac{14 + \sqrt{\frac{5}{3}} \cdot 125 - 11 \cdot 5}{125}$

Step 3

Simplify each term.

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Step 3.1

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

$\frac{14 + \frac{\sqrt{5}}{\sqrt{3}} \cdot 125 - 11 \cdot 5}{125}$

Step 3.2

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

$\frac{14 + \frac{\sqrt{5}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \cdot 125 - 11 \cdot 5}{125}$

Step 3.3

Combine and simplify the denominator.

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Step 3.3.1

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

$\frac{14 + \frac{\sqrt{5} \sqrt{3}}{\sqrt{3} \sqrt{3}} \cdot 125 - 11 \cdot 5}{125}$

Step 3.3.2

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

$\frac{14 + \frac{\sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} \cdot 125 - 11 \cdot 5}{125}$

Step 3.3.3

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

$\frac{14 + \frac{\sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} \cdot 125 - 11 \cdot 5}{125}$

Step 3.3.4

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

$\frac{14 + \frac{\sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1 + 1}} \cdot 125 - 11 \cdot 5}{125}$

Step 3.3.5

Add $1$ and $1$ .

$\frac{14 + \frac{\sqrt{5} \sqrt{3}}{{\sqrt{3}}^{2}} \cdot 125 - 11 \cdot 5}{125}$

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

$\frac{14 + \frac{\sqrt{5} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} \cdot 125 - 11 \cdot 5}{125}$

Step 3.3.6.2

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

$\frac{14 + \frac{\sqrt{5} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} \cdot 125 - 11 \cdot 5}{125}$

Step 3.3.6.3

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

$\frac{14 + \frac{\sqrt{5} \sqrt{3}}{3^{\frac{2}{2}}} \cdot 125 - 11 \cdot 5}{125}$

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.

$\frac{14 + \frac{\sqrt{5} \sqrt{3}}{3^{\frac{2}{2}}} \cdot 125 - 11 \cdot 5}{125}$

Step 3.3.6.4.2

Rewrite the expression.

$\frac{14 + \frac{\sqrt{5} \sqrt{3}}{3^{1}} \cdot 125 - 11 \cdot 5}{125}$

Step 3.3.6.5

Evaluate the exponent.

$\frac{14 + \frac{\sqrt{5} \sqrt{3}}{3} \cdot 125 - 11 \cdot 5}{125}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Combine using the product rule for radicals.

$\frac{14 + \frac{\sqrt{5 \cdot 3}}{3} \cdot 125 - 11 \cdot 5}{125}$

Step 3.4.2

Multiply $5$ by $3$ .

$\frac{14 + \frac{\sqrt{15}}{3} \cdot 125 - 11 \cdot 5}{125}$

Step 3.5

Combine $\frac{\sqrt{15}}{3}$ and $125$ .

$\frac{14 + \frac{\sqrt{15} \cdot 125}{3} - 11 \cdot 5}{125}$

Step 3.6

Move $125$ to the left of $\sqrt{15}$ .

$\frac{14 + \frac{125 \sqrt{15}}{3} - 11 \cdot 5}{125}$

Step 3.7

Multiply $- 11$ by $5$ .

$\frac{14 + \frac{125 \sqrt{15}}{3} - 55}{125}$

Step 4

Find the common denominator.

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Step 4.1

Write $14$ as a fraction with denominator $1$ .

$\frac{\frac{14}{1} + \frac{125 \sqrt{15}}{3} - 55}{125}$

Step 4.2

Multiply $\frac{14}{1}$ by $\frac{3}{3}$ .

$\frac{\frac{14}{1} \cdot \frac{3}{3} + \frac{125 \sqrt{15}}{3} - 55}{125}$

Step 4.3

Multiply $\frac{14}{1}$ by $\frac{3}{3}$ .

$\frac{\frac{14 \cdot 3}{3} + \frac{125 \sqrt{15}}{3} - 55}{125}$

Step 4.4

Write $- 55$ as a fraction with denominator $1$ .

$\frac{\frac{14 \cdot 3}{3} + \frac{125 \sqrt{15}}{3} + \frac{- 55}{1}}{125}$

Step 4.5

Multiply $\frac{- 55}{1}$ by $\frac{3}{3}$ .

$\frac{\frac{14 \cdot 3}{3} + \frac{125 \sqrt{15}}{3} + \frac{- 55}{1} \cdot \frac{3}{3}}{125}$

Step 4.6

Multiply $\frac{- 55}{1}$ by $\frac{3}{3}$ .

$\frac{\frac{14 \cdot 3}{3} + \frac{125 \sqrt{15}}{3} + \frac{- 55 \cdot 3}{3}}{125}$

Step 5

Combine the numerators over the common denominator.

$\frac{\frac{14 \cdot 3 + 125 \sqrt{15} - 55 \cdot 3}{3}}{125}$

Step 6

Simplify each term.

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Step 6.1

Multiply $14$ by $3$ .

$\frac{\frac{42 + 125 \sqrt{15} - 55 \cdot 3}{3}}{125}$

Step 6.2

Multiply $- 55$ by $3$ .

$\frac{\frac{42 + 125 \sqrt{15} - 165}{3}}{125}$

Step 7

Simplify with factoring out.

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Step 7.1

Subtract $165$ from $42$ .

$\frac{\frac{- 123 + 125 \sqrt{15}}{3}}{125}$

Step 7.2

Rewrite $- 123$ as $- 1 (123)$ .

$\frac{\frac{- 1 (123) + 125 \sqrt{15}}{3}}{125}$

Step 7.3

Factor $- 1$ out of $125 \sqrt{15}$ .

$\frac{\frac{- 1 (123) - (- 125 \sqrt{15})}{3}}{125}$

Step 7.4

Factor $- 1$ out of $- 1 (123) - (- 125 \sqrt{15})$ .

$\frac{\frac{- 1 (123 - 125 \sqrt{15})}{3}}{125}$

Step 7.5

Move the negative in front of the fraction.

$\frac{- \frac{123 - 125 \sqrt{15}}{3}}{125}$

Step 8

Multiply the numerator by the reciprocal of the denominator.

$- \frac{123 - 125 \sqrt{15}}{3} \cdot \frac{1}{125}$

Step 9

Multiply $- \frac{123 - 125 \sqrt{15}}{3} \cdot \frac{1}{125}$ .

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Step 9.1

Multiply $\frac{1}{125}$ by $\frac{123 - 125 \sqrt{15}}{3}$ .

$- \frac{123 - 125 \sqrt{15}}{125 \cdot 3}$

Step 9.2

Multiply $125$ by $3$ .

$- \frac{123 - 125 \sqrt{15}}{375}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{123 - 125 \sqrt{15}}{375}$

Decimal Form:

$0.96299444 \dots$