Finite Math Examples

$\sqrt[3]{\frac{125^{n} \cdot 5^{4 n}}{25^{- n}}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $125$ as $5^{3}$ .

$\sqrt[3]{\frac{{(5^{3})}^{n} \cdot 5^{4 n}}{25^{- n}}}$

Step 1.2

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

$\sqrt[3]{\frac{5^{3 n} \cdot 5^{4 n}}{25^{- n}}}$

Step 1.3

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

$\sqrt[3]{\frac{5^{3 n + 4 n}}{25^{- n}}}$

Step 1.4

Add $3 n$ and $4 n$ .

$\sqrt[3]{\frac{5^{7 n}}{25^{- n}}}$

Step 2

Rewrite $\sqrt[3]{\frac{5^{7 n}}{25^{- n}}}$ as $\frac{\sqrt[3]{5^{7 n}}}{\sqrt[3]{25^{- n}}}$ .

$\frac{\sqrt[3]{5^{7 n}}}{\sqrt[3]{25^{- n}}}$

Step 3

Multiply $\frac{\sqrt[3]{5^{7 n}}}{\sqrt[3]{25^{- n}}}$ by $\frac{{\sqrt[3]{25^{- n}}}^{2}}{{\sqrt[3]{25^{- n}}}^{2}}$ .

$\frac{\sqrt[3]{5^{7 n}}}{\sqrt[3]{25^{- n}}} \cdot \frac{{\sqrt[3]{25^{- n}}}^{2}}{{\sqrt[3]{25^{- n}}}^{2}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{\sqrt[3]{5^{7 n}}}{\sqrt[3]{25^{- n}}}$ by $\frac{{\sqrt[3]{25^{- n}}}^{2}}{{\sqrt[3]{25^{- n}}}^{2}}$ .

$\frac{\sqrt[3]{5^{7 n}} {\sqrt[3]{25^{- n}}}^{2}}{\sqrt[3]{25^{- n}} {\sqrt[3]{25^{- n}}}^{2}}$

Step 4.2

Raise $\sqrt[3]{25^{- n}}$ to the power of $1$ .

$\frac{\sqrt[3]{5^{7 n}} {\sqrt[3]{25^{- n}}}^{2}}{{\sqrt[3]{25^{- n}}}^{1} {\sqrt[3]{25^{- n}}}^{2}}$

Step 4.3

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

$\frac{\sqrt[3]{5^{7 n}} {\sqrt[3]{25^{- n}}}^{2}}{{\sqrt[3]{25^{- n}}}^{1 + 2}}$

Step 4.4

Add $1$ and $2$ .

$\frac{\sqrt[3]{5^{7 n}} {\sqrt[3]{25^{- n}}}^{2}}{{\sqrt[3]{25^{- n}}}^{3}}$

Step 4.5

Rewrite ${\sqrt[3]{25^{- n}}}^{3}$ as $25^{- n}$ .

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Step 4.5.1

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

$\frac{\sqrt[3]{5^{7 n}} {\sqrt[3]{25^{- n}}}^{2}}{{(25^{\frac{- n}{3}})}^{3}}$

Step 4.5.2

Move the negative in front of the fraction.

$\frac{\sqrt[3]{5^{7 n}} {\sqrt[3]{25^{- n}}}^{2}}{{(25^{- \frac{n}{3}})}^{3}}$

Step 4.5.3

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

$\frac{\sqrt[3]{5^{7 n}} {\sqrt[3]{25^{- n}}}^{2}}{25^{- \frac{n}{3} \cdot 3}}$

Step 4.5.4

Multiply $3$ by $- 1$ .

$\frac{\sqrt[3]{5^{7 n}} {\sqrt[3]{25^{- n}}}^{2}}{25^{- 3 \frac{n}{3}}}$

Step 4.5.5

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

$\frac{\sqrt[3]{5^{7 n}} {\sqrt[3]{25^{- n}}}^{2}}{25^{\frac{- 3 n}{3}}}$

Step 4.5.6

Cancel the common factor of $- 3$ and $3$ .

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Step 4.5.6.1

Factor $3$ out of $- 3 n$ .

$\frac{\sqrt[3]{5^{7 n}} {\sqrt[3]{25^{- n}}}^{2}}{25^{\frac{3 (- n)}{3}}}$

Step 4.5.6.2

Cancel the common factors.

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Step 4.5.6.2.1

Factor $3$ out of $3$ .

$\frac{\sqrt[3]{5^{7 n}} {\sqrt[3]{25^{- n}}}^{2}}{25^{\frac{3 (- n)}{3 (1)}}}$

Step 4.5.6.2.2

Cancel the common factor.

$\frac{\sqrt[3]{5^{7 n}} {\sqrt[3]{25^{- n}}}^{2}}{25^{\frac{3 (- n)}{3 \cdot 1}}}$

Step 4.5.6.2.3

Rewrite the expression.

$\frac{\sqrt[3]{5^{7 n}} {\sqrt[3]{25^{- n}}}^{2}}{25^{\frac{- n}{1}}}$

Step 4.5.6.2.4

Divide $- n$ by $1$ .

$\frac{\sqrt[3]{5^{7 n}} {\sqrt[3]{25^{- n}}}^{2}}{25^{- n}}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite ${\sqrt[3]{25^{- n}}}^{2}$ as $\sqrt[3]{{(25^{- n})}^{2}}$ .

$\frac{\sqrt[3]{5^{7 n}} \sqrt[3]{{(25^{- n})}^{2}}}{25^{- n}}$

Step 5.2

Multiply the exponents in ${(25^{- n})}^{2}$ .

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Step 5.2.1

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

$\frac{\sqrt[3]{5^{7 n}} \sqrt[3]{25^{- n \cdot 2}}}{25^{- n}}$

Step 5.2.2

Multiply $2$ by $- 1$ .

$\frac{\sqrt[3]{5^{7 n}} \sqrt[3]{25^{- 2 n}}}{25^{- n}}$

Step 6

Simplify the numerator.

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Step 6.1

Combine using the product rule for radicals.

$\frac{\sqrt[3]{5^{7 n} \cdot 25^{- 2 n}}}{25^{- n}}$

Step 6.2

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

$\frac{\sqrt[3]{5^{7 n} \cdot {(5^{2})}^{- 2 n}}}{25^{- n}}$

Step 6.3

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

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Step 6.3.1

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

$\frac{\sqrt[3]{5^{7 n} \cdot 5^{2 (- 2 n)}}}{25^{- n}}$

Step 6.3.2

Multiply $- 2$ by $2$ .

$\frac{\sqrt[3]{5^{7 n} \cdot 5^{- 4 n}}}{25^{- n}}$

Step 6.4

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

$\frac{\sqrt[3]{5^{7 n - 4 n}}}{25^{- n}}$

Step 6.5

Subtract $4 n$ from $7 n$ .

$\frac{\sqrt[3]{5^{3 n}}}{25^{- n}}$

Step 7

Simplify the numerator.

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Step 7.1

Rewrite $5^{3 n}$ as ${(5^{n})}^{3}$ .

$\frac{\sqrt[3]{{(5^{n})}^{3}}}{25^{- n}}$

Step 7.2

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

$\frac{5^{n}}{25^{- n}}$