Basic Math Examples

$\frac{3 + \sqrt[3]{\frac{1}{5}}}{\frac{13}{5} + \sqrt{\frac{1}{125}}}$

Step 1

Multiply the numerator and denominator of the fraction by $5$ .

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Step 1.1

Multiply $\frac{3 + \sqrt[3]{\frac{1}{5}}}{\frac{13}{5} + \sqrt{\frac{1}{125}}}$ by $\frac{5}{5}$ .

$\frac{5}{5} \cdot \frac{3 + \sqrt[3]{\frac{1}{5}}}{\frac{13}{5} + \sqrt{\frac{1}{125}}}$

Step 1.2

Combine.

$\frac{5 (3 + \sqrt[3]{\frac{1}{5}})}{5 (\frac{13}{5} + \sqrt{\frac{1}{125}})}$

Step 2

Apply the distributive property.

$\frac{5 \cdot 3 + 5 \sqrt[3]{\frac{1}{5}}}{5 (\frac{13}{5}) + 5 \sqrt{\frac{1}{125}}}$

Step 3

Cancel the common factor of $5$ .

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Step 3.1

Cancel the common factor.

$\frac{5 \cdot 3 + 5 \sqrt[3]{\frac{1}{5}}}{5 (\frac{13}{5}) + 5 \sqrt{\frac{1}{125}}}$

Step 3.2

Rewrite the expression.

$\frac{5 \cdot 3 + 5 \sqrt[3]{\frac{1}{5}}}{13 + 5 \sqrt{\frac{1}{125}}}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $5$ by $3$ .

$\frac{15 + 5 \sqrt[3]{\frac{1}{5}}}{13 + 5 \sqrt{\frac{1}{125}}}$

Step 4.2

Rewrite $\sqrt[3]{\frac{1}{5}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{5}}$ .

$\frac{15 + 5 \frac{\sqrt[3]{1}}{\sqrt[3]{5}}}{13 + 5 \sqrt{\frac{1}{125}}}$

Step 4.3

Any root of $1$ is $1$ .

$\frac{15 + 5 \frac{1}{\sqrt[3]{5}}}{13 + 5 \sqrt{\frac{1}{125}}}$

Step 4.4

Multiply $\frac{1}{\sqrt[3]{5}}$ by $\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$ .

$\frac{15 + 5 (\frac{1}{\sqrt[3]{5}} \cdot \frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}})}{13 + 5 \sqrt{\frac{1}{125}}}$

Step 4.5

Combine and simplify the denominator.

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Step 4.5.1

Multiply $\frac{1}{\sqrt[3]{5}}$ by $\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$ .

$\frac{15 + 5 \frac{{\sqrt[3]{5}}^{2}}{\sqrt[3]{5} {\sqrt[3]{5}}^{2}}}{13 + 5 \sqrt{\frac{1}{125}}}$

Step 4.5.2

Raise $\sqrt[3]{5}$ to the power of $1$ .

$\frac{15 + 5 \frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{1} {\sqrt[3]{5}}^{2}}}{13 + 5 \sqrt{\frac{1}{125}}}$

Step 4.5.3

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

$\frac{15 + 5 \frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{1 + 2}}}{13 + 5 \sqrt{\frac{1}{125}}}$

Step 4.5.4

Add $1$ and $2$ .

$\frac{15 + 5 \frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{3}}}{13 + 5 \sqrt{\frac{1}{125}}}$

Step 4.5.5

Rewrite ${\sqrt[3]{5}}^{3}$ as $5$ .

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Step 4.5.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5}$ as $5^{\frac{1}{3}}$ .

$\frac{15 + 5 \frac{{\sqrt[3]{5}}^{2}}{{(5^{\frac{1}{3}})}^{3}}}{13 + 5 \sqrt{\frac{1}{125}}}$

Step 4.5.5.2

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

$\frac{15 + 5 \frac{{\sqrt[3]{5}}^{2}}{5^{\frac{1}{3} \cdot 3}}}{13 + 5 \sqrt{\frac{1}{125}}}$

Step 4.5.5.3

Combine $\frac{1}{3}$ and $3$ .

$\frac{15 + 5 \frac{{\sqrt[3]{5}}^{2}}{5^{\frac{3}{3}}}}{13 + 5 \sqrt{\frac{1}{125}}}$

Step 4.5.5.4

Cancel the common factor of $3$ .

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Step 4.5.5.4.1

Cancel the common factor.

$\frac{15 + 5 \frac{{\sqrt[3]{5}}^{2}}{5^{\frac{3}{3}}}}{13 + 5 \sqrt{\frac{1}{125}}}$

Step 4.5.5.4.2

Rewrite the expression.

$\frac{15 + 5 \frac{{\sqrt[3]{5}}^{2}}{5^{1}}}{13 + 5 \sqrt{\frac{1}{125}}}$

Step 4.5.5.5

Evaluate the exponent.

$\frac{15 + 5 \frac{{\sqrt[3]{5}}^{2}}{5}}{13 + 5 \sqrt{\frac{1}{125}}}$

Step 4.6

Cancel the common factor of $5$ .

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Step 4.6.1

Cancel the common factor.

$\frac{15 + 5 \frac{{\sqrt[3]{5}}^{2}}{5}}{13 + 5 \sqrt{\frac{1}{125}}}$

Step 4.6.2

Rewrite the expression.

$\frac{15 + {\sqrt[3]{5}}^{2}}{13 + 5 \sqrt{\frac{1}{125}}}$

Step 4.7

Rewrite ${\sqrt[3]{5}}^{2}$ as $\sqrt[3]{5^{2}}$ .

$\frac{15 + \sqrt[3]{5^{2}}}{13 + 5 \sqrt{\frac{1}{125}}}$

Step 4.8

Raise $5$ to the power of $2$ .

$\frac{15 + \sqrt[3]{25}}{13 + 5 \sqrt{\frac{1}{125}}}$

Step 5

Simplify the denominator.

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Step 5.1

Rewrite $\sqrt{\frac{1}{125}}$ as $\frac{\sqrt{1}}{\sqrt{125}}$ .

$\frac{15 + \sqrt[3]{25}}{13 + 5 \frac{\sqrt{1}}{\sqrt{125}}}$

Step 5.2

Any root of $1$ is $1$ .

$\frac{15 + \sqrt[3]{25}}{13 + 5 \frac{1}{\sqrt{125}}}$

Step 5.3

Simplify the denominator.

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Step 5.3.1

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 5.3.1.1

Factor $25$ out of $125$ .

$\frac{15 + \sqrt[3]{25}}{13 + 5 \frac{1}{\sqrt{25 (5)}}}$

Step 5.3.1.2

Rewrite $25$ as $5^{2}$ .

$\frac{15 + \sqrt[3]{25}}{13 + 5 \frac{1}{\sqrt{5^{2} \cdot 5}}}$

Step 5.3.2

Pull terms out from under the radical.

$\frac{15 + \sqrt[3]{25}}{13 + 5 \frac{1}{5 \sqrt{5}}}$

Step 5.4

Cancel the common factor of $5$ .

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Step 5.4.1

Factor $5$ out of $5 \sqrt{5}$ .

$\frac{15 + \sqrt[3]{25}}{13 + 5 \frac{1}{5 (\sqrt{5})}}$

Step 5.4.2

Cancel the common factor.

$\frac{15 + \sqrt[3]{25}}{13 + 5 \frac{1}{5 \sqrt{5}}}$

Step 5.4.3

Rewrite the expression.

$\frac{15 + \sqrt[3]{25}}{13 + \frac{1}{\sqrt{5}}}$

Step 5.5

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

$\frac{15 + \sqrt[3]{25}}{13 + \frac{1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}}$

Step 5.6

Combine and simplify the denominator.

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Step 5.6.1

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

$\frac{15 + \sqrt[3]{25}}{13 + \frac{\sqrt{5}}{\sqrt{5} \sqrt{5}}}$

Step 5.6.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{15 + \sqrt[3]{25}}{13 + \frac{\sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}}$

Step 5.6.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{15 + \sqrt[3]{25}}{13 + \frac{\sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}}$

Step 5.6.4

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

$\frac{15 + \sqrt[3]{25}}{13 + \frac{\sqrt{5}}{{\sqrt{5}}^{1 + 1}}}$

Step 5.6.5

Add $1$ and $1$ .

$\frac{15 + \sqrt[3]{25}}{13 + \frac{\sqrt{5}}{{\sqrt{5}}^{2}}}$

Step 5.6.6

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

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Step 5.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{15 + \sqrt[3]{25}}{13 + \frac{\sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}}$

Step 5.6.6.2

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

$\frac{15 + \sqrt[3]{25}}{13 + \frac{\sqrt{5}}{5^{\frac{1}{2} \cdot 2}}}$

Step 5.6.6.3

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

$\frac{15 + \sqrt[3]{25}}{13 + \frac{\sqrt{5}}{5^{\frac{2}{2}}}}$

Step 5.6.6.4

Cancel the common factor of $2$ .

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Step 5.6.6.4.1

Cancel the common factor.

$\frac{15 + \sqrt[3]{25}}{13 + \frac{\sqrt{5}}{5^{\frac{2}{2}}}}$

Step 5.6.6.4.2

Rewrite the expression.

$\frac{15 + \sqrt[3]{25}}{13 + \frac{\sqrt{5}}{5^{1}}}$

Step 5.6.6.5

Evaluate the exponent.

$\frac{15 + \sqrt[3]{25}}{13 + \frac{\sqrt{5}}{5}}$

Step 5.7

To write $13$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{15 + \sqrt[3]{25}}{13 \cdot \frac{5}{5} + \frac{\sqrt{5}}{5}}$

Step 5.8

Combine $13$ and $\frac{5}{5}$ .

$\frac{15 + \sqrt[3]{25}}{\frac{13 \cdot 5}{5} + \frac{\sqrt{5}}{5}}$

Step 5.9

Combine the numerators over the common denominator.

$\frac{15 + \sqrt[3]{25}}{\frac{13 \cdot 5 + \sqrt{5}}{5}}$

Step 5.10

Multiply $13$ by $5$ .

$\frac{15 + \sqrt[3]{25}}{\frac{65 + \sqrt{5}}{5}}$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$(15 + \sqrt[3]{25}) \frac{5}{65 + \sqrt{5}}$

Step 7

Multiply $\frac{5}{65 + \sqrt{5}}$ by $\frac{65 - \sqrt{5}}{65 - \sqrt{5}}$ .

$(15 + \sqrt[3]{25}) (\frac{5}{65 + \sqrt{5}} \cdot \frac{65 - \sqrt{5}}{65 - \sqrt{5}})$

Step 8

Multiply $\frac{5}{65 + \sqrt{5}}$ by $\frac{65 - \sqrt{5}}{65 - \sqrt{5}}$ .

$(15 + \sqrt[3]{25}) \frac{5 (65 - \sqrt{5})}{(65 + \sqrt{5}) (65 - \sqrt{5})}$

Step 9

Expand the denominator using the FOIL method.

$(15 + \sqrt[3]{25}) \frac{5 (65 - \sqrt{5})}{4225 - 65 \sqrt{5} + \sqrt{5} \cdot 65 - {\sqrt{5}}^{2}}$

Step 10

Simplify.

$(15 + \sqrt[3]{25}) \frac{5 (65 - \sqrt{5})}{4220}$

Step 11

Cancel the common factors.

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Step 11.1

Factor $5$ out of $4220$ .

$(15 + \sqrt[3]{25}) \frac{5 (65 - \sqrt{5})}{5 \cdot 844}$

Step 11.2

Cancel the common factor.

$(15 + \sqrt[3]{25}) \frac{5 (65 - \sqrt{5})}{5 \cdot 844}$

Step 11.3

Rewrite the expression.

$(15 + \sqrt[3]{25}) \frac{65 - \sqrt{5}}{844}$

Step 12

Simplify terms.

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Step 12.1

Apply the distributive property.

$15 \frac{65 - \sqrt{5}}{844} + \sqrt[3]{25} \frac{65 - \sqrt{5}}{844}$

Step 12.2

Combine $15$ and $\frac{65 - \sqrt{5}}{844}$ .

$\frac{15 (65 - \sqrt{5})}{844} + \sqrt[3]{25} \frac{65 - \sqrt{5}}{844}$

Step 12.3

Combine $\sqrt[3]{25}$ and $\frac{65 - \sqrt{5}}{844}$ .

$\frac{15 (65 - \sqrt{5})}{844} + \frac{\sqrt[3]{25} (65 - \sqrt{5})}{844}$

Step 12.4

Combine the numerators over the common denominator.

$\frac{15 (65 - \sqrt{5}) + \sqrt[3]{25} (65 - \sqrt{5})}{844}$

Step 13

Simplify the numerator.

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Step 13.1

Apply the distributive property.

$\frac{15 \cdot 65 + 15 (- \sqrt{5}) + \sqrt[3]{25} (65 - \sqrt{5})}{844}$

Step 13.2

Multiply $15$ by $65$ .

$\frac{975 + 15 (- \sqrt{5}) + \sqrt[3]{25} (65 - \sqrt{5})}{844}$

Step 13.3

Multiply $- 1$ by $15$ .

$\frac{975 - 15 \sqrt{5} + \sqrt[3]{25} (65 - \sqrt{5})}{844}$

Step 13.4

Apply the distributive property.

$\frac{975 - 15 \sqrt{5} + \sqrt[3]{25} \cdot 65 + \sqrt[3]{25} (- \sqrt{5})}{844}$

Step 13.5

Move $65$ to the left of $\sqrt[3]{25}$ .

$\frac{975 - 15 \sqrt{5} + 65 \cdot \sqrt[3]{25} + \sqrt[3]{25} (- \sqrt{5})}{844}$

Step 13.6

Multiply $\sqrt[3]{25} (- \sqrt{5})$ .

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Step 13.6.1

Rewrite the expression using the least common index of $6$ .

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Step 13.6.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{25}$ as $25^{\frac{1}{3}}$ .

$\frac{975 - 15 \sqrt{5} + 65 \cdot \sqrt[3]{25} - (25^{\frac{1}{3}} \sqrt{5})}{844}$

Step 13.6.1.2

Rewrite $25^{\frac{1}{3}}$ as $25^{\frac{2}{6}}$ .

$\frac{975 - 15 \sqrt{5} + 65 \cdot \sqrt[3]{25} - (25^{\frac{2}{6}} \sqrt{5})}{844}$

Step 13.6.1.3

Rewrite $25^{\frac{2}{6}}$ as $\sqrt[6]{25^{2}}$ .

$\frac{975 - 15 \sqrt{5} + 65 \cdot \sqrt[3]{25} - (\sqrt[6]{25^{2}} \sqrt{5})}{844}$

Step 13.6.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{975 - 15 \sqrt{5} + 65 \cdot \sqrt[3]{25} - (\sqrt[6]{25^{2}} \cdot 5^{\frac{1}{2}})}{844}$

Step 13.6.1.5

Rewrite $5^{\frac{1}{2}}$ as $5^{\frac{3}{6}}$ .

$\frac{975 - 15 \sqrt{5} + 65 \cdot \sqrt[3]{25} - (\sqrt[6]{25^{2}} \cdot 5^{\frac{3}{6}})}{844}$

Step 13.6.1.6

Rewrite $5^{\frac{3}{6}}$ as $\sqrt[6]{5^{3}}$ .

$\frac{975 - 15 \sqrt{5} + 65 \cdot \sqrt[3]{25} - (\sqrt[6]{25^{2}} \sqrt[6]{5^{3}})}{844}$

Step 13.6.2

Combine using the product rule for radicals.

$\frac{975 - 15 \sqrt{5} + 65 \cdot \sqrt[3]{25} - \sqrt[6]{25^{2} \cdot 5^{3}}}{844}$

Step 13.6.3

Rewrite $25$ as $5^{2}$ .

$\frac{975 - 15 \sqrt{5} + 65 \cdot \sqrt[3]{25} - \sqrt[6]{{(5^{2})}^{2} \cdot 5^{3}}}{844}$

Step 13.6.4

Multiply the exponents in ${(5^{2})}^{2}$ .

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Step 13.6.4.1

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

$\frac{975 - 15 \sqrt{5} + 65 \cdot \sqrt[3]{25} - \sqrt[6]{5^{2 \cdot 2} \cdot 5^{3}}}{844}$

Step 13.6.4.2

Multiply $2$ by $2$ .

$\frac{975 - 15 \sqrt{5} + 65 \cdot \sqrt[3]{25} - \sqrt[6]{5^{4} \cdot 5^{3}}}{844}$

Step 13.6.5

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

$\frac{975 - 15 \sqrt{5} + 65 \cdot \sqrt[3]{25} - \sqrt[6]{5^{4 + 3}}}{844}$

Step 13.6.6

Add $4$ and $3$ .

$\frac{975 - 15 \sqrt{5} + 65 \cdot \sqrt[3]{25} - \sqrt[6]{5^{7}}}{844}$

Step 13.7

Simplify each term.

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Step 13.7.1

Raise $5$ to the power of $7$ .

$\frac{975 - 15 \sqrt{5} + 65 \sqrt[3]{25} - \sqrt[6]{78125}}{844}$

Step 13.7.2

Rewrite $78125$ as $5^{6} \cdot 5$ .

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Step 13.7.2.1

Factor $15625$ out of $78125$ .

$\frac{975 - 15 \sqrt{5} + 65 \sqrt[3]{25} - \sqrt[6]{15625 (5)}}{844}$

Step 13.7.2.2

Rewrite $15625$ as $5^{6}$ .

$\frac{975 - 15 \sqrt{5} + 65 \sqrt[3]{25} - \sqrt[6]{5^{6} \cdot 5}}{844}$

Step 13.7.3

Pull terms out from under the radical.

$\frac{975 - 15 \sqrt{5} + 65 \sqrt[3]{25} - (5 \sqrt[6]{5})}{844}$

Step 13.7.4

Multiply $5$ by $- 1$ .

$\frac{975 - 15 \sqrt{5} + 65 \sqrt[3]{25} - 5 \sqrt[6]{5}}{844}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{975 - 15 \sqrt{5} + 65 \sqrt[3]{25} - 5 \sqrt[6]{5}}{844}$

Decimal Form:

$1.33291686 \dots$