Finite Math Examples

$\frac{45}{\sqrt[5]{125 x^{3}}}$

Step 1

Multiply $\frac{45}{\sqrt[5]{125 x^{3}}}$ by $\frac{{\sqrt[5]{125 x^{3}}}^{4}}{{\sqrt[5]{125 x^{3}}}^{4}}$ .

$\frac{45}{\sqrt[5]{125 x^{3}}} \cdot \frac{{\sqrt[5]{125 x^{3}}}^{4}}{{\sqrt[5]{125 x^{3}}}^{4}}$

Step 2

Simplify terms.

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Step 2.1

Combine and simplify the denominator.

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Step 2.1.1

Multiply $\frac{45}{\sqrt[5]{125 x^{3}}}$ by $\frac{{\sqrt[5]{125 x^{3}}}^{4}}{{\sqrt[5]{125 x^{3}}}^{4}}$ .

$\frac{45 {\sqrt[5]{125 x^{3}}}^{4}}{\sqrt[5]{125 x^{3}} {\sqrt[5]{125 x^{3}}}^{4}}$

Step 2.1.2

Raise $\sqrt[5]{125 x^{3}}$ to the power of $1$ .

$\frac{45 {\sqrt[5]{125 x^{3}}}^{4}}{{\sqrt[5]{125 x^{3}}}^{1} {\sqrt[5]{125 x^{3}}}^{4}}$

Step 2.1.3

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

$\frac{45 {\sqrt[5]{125 x^{3}}}^{4}}{{\sqrt[5]{125 x^{3}}}^{1 + 4}}$

Step 2.1.4

Add $1$ and $4$ .

$\frac{45 {\sqrt[5]{125 x^{3}}}^{4}}{{\sqrt[5]{125 x^{3}}}^{5}}$

Step 2.1.5

Rewrite ${\sqrt[5]{125 x^{3}}}^{5}$ as $125 x^{3}$ .

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Step 2.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{125 x^{3}}$ as ${(125 x^{3})}^{\frac{1}{5}}$ .

$\frac{45 {\sqrt[5]{125 x^{3}}}^{4}}{{({(125 x^{3})}^{\frac{1}{5}})}^{5}}$

Step 2.1.5.2

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

$\frac{45 {\sqrt[5]{125 x^{3}}}^{4}}{{(125 x^{3})}^{\frac{1}{5} \cdot 5}}$

Step 2.1.5.3

Combine $\frac{1}{5}$ and $5$ .

$\frac{45 {\sqrt[5]{125 x^{3}}}^{4}}{{(125 x^{3})}^{\frac{5}{5}}}$

Step 2.1.5.4

Cancel the common factor of $5$ .

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Step 2.1.5.4.1

Cancel the common factor.

$\frac{45 {\sqrt[5]{125 x^{3}}}^{4}}{{(125 x^{3})}^{\frac{5}{5}}}$

Step 2.1.5.4.2

Rewrite the expression.

$\frac{45 {\sqrt[5]{125 x^{3}}}^{4}}{{(125 x^{3})}^{1}}$

Step 2.1.5.5

Simplify.

$\frac{45 {\sqrt[5]{125 x^{3}}}^{4}}{125 x^{3}}$

Step 2.2

Cancel the common factor of $45$ and $125$ .

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Step 2.2.1

Factor $5$ out of $45 {\sqrt[5]{125 x^{3}}}^{4}$ .

$\frac{5 (9 {\sqrt[5]{125 x^{3}}}^{4})}{125 x^{3}}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Factor $5$ out of $125 x^{3}$ .

$\frac{5 (9 {\sqrt[5]{125 x^{3}}}^{4})}{5 (25 x^{3})}$

Step 2.2.2.2

Cancel the common factor.

$\frac{5 (9 {\sqrt[5]{125 x^{3}}}^{4})}{5 (25 x^{3})}$

Step 2.2.2.3

Rewrite the expression.

$\frac{9 {\sqrt[5]{125 x^{3}}}^{4}}{25 x^{3}}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite ${\sqrt[5]{125 x^{3}}}^{4}$ as $\sqrt[5]{{(125 x^{3})}^{4}}$ .

$\frac{9 \sqrt[5]{{(125 x^{3})}^{4}}}{25 x^{3}}$

Step 3.2

Apply the product rule to $125 x^{3}$ .

$\frac{9 \sqrt[5]{125^{4} {(x^{3})}^{4}}}{25 x^{3}}$

Step 3.3

Raise $125$ to the power of $4$ .

$\frac{9 \sqrt[5]{244140625 {(x^{3})}^{4}}}{25 x^{3}}$

Step 3.4

Multiply the exponents in ${(x^{3})}^{4}$ .

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Step 3.4.1

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

$\frac{9 \sqrt[5]{244140625 x^{3 \cdot 4}}}{25 x^{3}}$

Step 3.4.2

Multiply $3$ by $4$ .

$\frac{9 \sqrt[5]{244140625 x^{12}}}{25 x^{3}}$

Step 3.5

Rewrite $244140625 x^{12}$ as ${(25 x^{2})}^{5} \cdot (25 x^{2})$ .

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Step 3.5.1

Factor $9765625$ out of $244140625$ .

$\frac{9 \sqrt[5]{9765625 (25) x^{12}}}{25 x^{3}}$

Step 3.5.2

Rewrite $9765625$ as $25^{5}$ .

$\frac{9 \sqrt[5]{25^{5} \cdot 25 x^{12}}}{25 x^{3}}$

Step 3.5.3

Factor out $x^{10}$ .

$\frac{9 \sqrt[5]{25^{5} \cdot 25 (x^{10} x^{2})}}{25 x^{3}}$

Step 3.5.4

Rewrite $x^{10}$ as ${(x^{2})}^{5}$ .

$\frac{9 \sqrt[5]{25^{5} \cdot 25 ({(x^{2})}^{5} x^{2})}}{25 x^{3}}$

Step 3.5.5

Move $25$ .

$\frac{9 \sqrt[5]{25^{5} {(x^{2})}^{5} \cdot 25 x^{2}}}{25 x^{3}}$

Step 3.5.6

Rewrite $25^{5} {(x^{2})}^{5}$ as ${(25 x^{2})}^{5}$ .

$\frac{9 \sqrt[5]{{(25 x^{2})}^{5} \cdot 25 x^{2}}}{25 x^{3}}$

Step 3.5.7

Add parentheses.

$\frac{9 \sqrt[5]{{(25 x^{2})}^{5} \cdot (25 x^{2})}}{25 x^{3}}$

Step 3.6

Pull terms out from under the radical.

$\frac{9 \cdot 25 x^{2} \sqrt[5]{25 x^{2}}}{25 x^{3}}$

Step 3.7

Multiply $9$ by $25$ .

$\frac{225 x^{2} \sqrt[5]{25 x^{2}}}{25 x^{3}}$

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $225$ and $25$ .

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Step 4.1.1

Factor $25$ out of $225 x^{2} \sqrt[5]{25 x^{2}}$ .

$\frac{25 (9 x^{2} \sqrt[5]{25 x^{2}})}{25 x^{3}}$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $25$ out of $25 x^{3}$ .

$\frac{25 (9 x^{2} \sqrt[5]{25 x^{2}})}{25 (x^{3})}$

Step 4.1.2.2

Cancel the common factor.

$\frac{25 (9 x^{2} \sqrt[5]{25 x^{2}})}{25 x^{3}}$

Step 4.1.2.3

Rewrite the expression.

$\frac{9 x^{2} \sqrt[5]{25 x^{2}}}{x^{3}}$

Step 4.2

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 4.2.1

Factor $x^{2}$ out of $9 x^{2} \sqrt[5]{25 x^{2}}$ .

$\frac{x^{2} (9 \sqrt[5]{25 x^{2}})}{x^{3}}$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $x^{2}$ out of $x^{3}$ .

$\frac{x^{2} (9 \sqrt[5]{25 x^{2}})}{x^{2} x}$

Step 4.2.2.2

Cancel the common factor.

$\frac{x^{2} (9 \sqrt[5]{25 x^{2}})}{x^{2} x}$

Step 4.2.2.3

Rewrite the expression.

$\frac{9 \sqrt[5]{25 x^{2}}}{x}$