Finite Math Examples

$\frac{{(3 x^{- 3} \sqrt{y})}^{4} {(2 \sqrt{x} y)}^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1

Simplify the numerator.

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Step 1.1

Apply the product rule to $3 x^{- 3} \sqrt{y}$ .

$\frac{{(3 x^{- 3})}^{4} {\sqrt{y}}^{4} {(2 \sqrt{x} y)}^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.2

Apply the product rule to $2 \sqrt{x} y$ .

$\frac{{(3 x^{- 3})}^{4} {\sqrt{y}}^{4} {(2 \sqrt{x})}^{2} y^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{{(3 \frac{1}{x^{3}})}^{4} {\sqrt{y}}^{4} {(2 \sqrt{x})}^{2} y^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.4

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

$\frac{{(\frac{3}{x^{3}})}^{4} {\sqrt{y}}^{4} {(2 \sqrt{x})}^{2} y^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.5

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

$\frac{\frac{3^{4}}{{(x^{3})}^{4}} {\sqrt{y}}^{4} {(2 \sqrt{x})}^{2} y^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.6

Raise $3$ to the power of $4$ .

$\frac{\frac{81}{{(x^{3})}^{4}} {\sqrt{y}}^{4} {(2 \sqrt{x})}^{2} y^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.7

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

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Step 1.7.1

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

$\frac{\frac{81}{x^{3 \cdot 4}} {\sqrt{y}}^{4} {(2 \sqrt{x})}^{2} y^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.7.2

Multiply $3$ by $4$ .

$\frac{\frac{81}{x^{12}} {\sqrt{y}}^{4} {(2 \sqrt{x})}^{2} y^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.8

Rewrite ${\sqrt{y}}^{4}$ as $y^{2}$ .

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Step 1.8.1

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

$\frac{\frac{81}{x^{12}} {(y^{\frac{1}{2}})}^{4} {(2 \sqrt{x})}^{2} y^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.8.2

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

$\frac{\frac{81}{x^{12}} y^{\frac{1}{2} \cdot 4} {(2 \sqrt{x})}^{2} y^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.8.3

Combine $\frac{1}{2}$ and $4$ .

$\frac{\frac{81}{x^{12}} y^{\frac{4}{2}} {(2 \sqrt{x})}^{2} y^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.8.4

Cancel the common factor of $4$ and $2$ .

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Step 1.8.4.1

Factor $2$ out of $4$ .

$\frac{\frac{81}{x^{12}} y^{\frac{2 \cdot 2}{2}} {(2 \sqrt{x})}^{2} y^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.8.4.2

Cancel the common factors.

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Step 1.8.4.2.1

Factor $2$ out of $2$ .

$\frac{\frac{81}{x^{12}} y^{\frac{2 \cdot 2}{2 (1)}} {(2 \sqrt{x})}^{2} y^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.8.4.2.2

Cancel the common factor.

$\frac{\frac{81}{x^{12}} y^{\frac{2 \cdot 2}{2 \cdot 1}} {(2 \sqrt{x})}^{2} y^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.8.4.2.3

Rewrite the expression.

$\frac{\frac{81}{x^{12}} y^{\frac{2}{1}} {(2 \sqrt{x})}^{2} y^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.8.4.2.4

Divide $2$ by $1$ .

$\frac{\frac{81}{x^{12}} y^{2} {(2 \sqrt{x})}^{2} y^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.9

Apply the product rule to $2 \sqrt{x}$ .

$\frac{\frac{81}{x^{12}} y^{2} (2^{2} {\sqrt{x}}^{2}) y^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.10

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

$\frac{\frac{81}{x^{12}} y^{2} (4 {\sqrt{x}}^{2}) y^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.11

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

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Step 1.11.1

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

$\frac{\frac{81}{x^{12}} y^{2} (4 {(x^{\frac{1}{2}})}^{2}) y^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.11.2

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

$\frac{\frac{81}{x^{12}} y^{2} (4 x^{\frac{1}{2} \cdot 2}) y^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.11.3

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

$\frac{\frac{81}{x^{12}} y^{2} (4 x^{\frac{2}{2}}) y^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.11.4

Cancel the common factor of $2$ .

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Step 1.11.4.1

Cancel the common factor.

$\frac{\frac{81}{x^{12}} y^{2} (4 x^{\frac{2}{2}}) y^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.11.4.2

Rewrite the expression.

$\frac{\frac{81}{x^{12}} y^{2} (4 x^{1}) y^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.11.5

Simplify.

$\frac{\frac{81}{x^{12}} y^{2} (4 x) y^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

$\frac{\frac{81}{x^{12}} y^{2} \cdot 4 x y^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.12

Combine exponents.

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Step 1.12.1

Combine $\frac{81}{x^{12}}$ and $y^{2}$ .

$\frac{\frac{81 y^{2}}{x^{12}} \cdot 4 x y^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.12.2

Combine $\frac{81 y^{2}}{x^{12}}$ and $4$ .

$\frac{\frac{81 y^{2} \cdot 4}{x^{12}} x y^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.12.3

Multiply $4$ by $81$ .

$\frac{\frac{324 y^{2}}{x^{12}} x y^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.12.4

Combine $\frac{324 y^{2}}{x^{12}}$ and $x$ .

$\frac{\frac{324 y^{2} x}{x^{12}} y^{2}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.12.5

Combine $\frac{324 y^{2} x}{x^{12}}$ and $y^{2}$ .

$\frac{\frac{324 y^{2} x y^{2}}{x^{12}}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.12.6

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

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Step 1.12.6.1

Move $y^{2}$ .

$\frac{\frac{324 (y^{2} y^{2}) x}{x^{12}}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.12.6.2

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

$\frac{\frac{324 y^{2 + 2} x}{x^{12}}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.12.6.3

Add $2$ and $2$ .

$\frac{\frac{324 y^{4} x}{x^{12}}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.13

Reduce the expression $\frac{324 y^{4} x}{x^{12}}$ by cancelling the common factors.

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Step 1.13.1

Factor $x$ out of $324 y^{4} x$ .

$\frac{\frac{x (324 y^{4})}{x^{12}}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.13.2

Factor $x$ out of $x^{12}$ .

$\frac{\frac{x (324 y^{4})}{x \cdot x^{11}}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.13.3

Cancel the common factor.

$\frac{\frac{x (324 y^{4})}{x \cdot x^{11}}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 1.13.4

Rewrite the expression.

$\frac{\frac{324 y^{4}}{x^{11}}}{{(9 x^{- 2} y^{3})}^{2}}$

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $9 x^{- 2} y^{3}$ .

$\frac{\frac{324 y^{4}}{x^{11}}}{{(9 x^{- 2})}^{2} {(y^{3})}^{2}}$

Step 2.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\frac{324 y^{4}}{x^{11}}}{{(9 \frac{1}{x^{2}})}^{2} {(y^{3})}^{2}}$

Step 2.3

Combine $9$ and $\frac{1}{x^{2}}$ .

$\frac{\frac{324 y^{4}}{x^{11}}}{{(\frac{9}{x^{2}})}^{2} {(y^{3})}^{2}}$

Step 2.4

Apply the product rule to $\frac{9}{x^{2}}$ .

$\frac{\frac{324 y^{4}}{x^{11}}}{\frac{9^{2}}{{(x^{2})}^{2}} {(y^{3})}^{2}}$

Step 2.5

Raise $9$ to the power of $2$ .

$\frac{\frac{324 y^{4}}{x^{11}}}{\frac{81}{{(x^{2})}^{2}} {(y^{3})}^{2}}$

Step 2.6

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

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Step 2.6.1

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

$\frac{\frac{324 y^{4}}{x^{11}}}{\frac{81}{x^{2 \cdot 2}} {(y^{3})}^{2}}$

Step 2.6.2

Multiply $2$ by $2$ .

$\frac{\frac{324 y^{4}}{x^{11}}}{\frac{81}{x^{4}} {(y^{3})}^{2}}$

Step 2.7

Multiply the exponents in ${(y^{3})}^{2}$ .

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Step 2.7.1

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

$\frac{\frac{324 y^{4}}{x^{11}}}{\frac{81}{x^{4}} y^{3 \cdot 2}}$

Step 2.7.2

Multiply $3$ by $2$ .

$\frac{\frac{324 y^{4}}{x^{11}}}{\frac{81}{x^{4}} y^{6}}$

Step 3

Combine $\frac{81}{x^{4}}$ and $y^{6}$ .

$\frac{\frac{324 y^{4}}{x^{11}}}{\frac{81 y^{6}}{x^{4}}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{324 y^{4}}{x^{11}} \cdot \frac{x^{4}}{81 y^{6}}$

Step 5

Combine.

$\frac{324 y^{4} x^{4}}{x^{11} (81 y^{6})}$

Step 6

Cancel the common factor of $324$ and $81$ .

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Step 6.1

Factor $81$ out of $324 y^{4} x^{4}$ .

$\frac{81 (4 y^{4} x^{4})}{x^{11} (81 y^{6})}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $81$ out of $x^{11} (81 y^{6})$ .

$\frac{81 (4 y^{4} x^{4})}{81 (x^{11} (y^{6}))}$

Step 6.2.2

Cancel the common factor.

$\frac{81 (4 y^{4} x^{4})}{81 (x^{11} (y^{6}))}$

Step 6.2.3

Rewrite the expression.

$\frac{4 y^{4} x^{4}}{x^{11} (y^{6})}$

Step 7

Cancel the common factor of $y^{4}$ and $y^{6}$ .

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Step 7.1

Factor $y^{4}$ out of $4 y^{4} x^{4}$ .

$\frac{y^{4} (4 x^{4})}{x^{11} (y^{6})}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $y^{4}$ out of $x^{11} (y^{6})$ .

$\frac{y^{4} (4 x^{4})}{y^{4} (x^{11} (y^{2}))}$

Step 7.2.2

Cancel the common factor.

$\frac{y^{4} (4 x^{4})}{y^{4} (x^{11} (y^{2}))}$

Step 7.2.3

Rewrite the expression.

$\frac{4 x^{4}}{x^{11} (y^{2})}$

Step 8

Cancel the common factor of $x^{4}$ and $x^{11}$ .

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Step 8.1

Factor $x^{4}$ out of $4 x^{4}$ .

$\frac{x^{4} \cdot 4}{x^{11} (y^{2})}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $x^{4}$ out of $x^{11} (y^{2})$ .

$\frac{x^{4} \cdot 4}{x^{4} (x^{7} y^{2})}$

Step 8.2.2

Cancel the common factor.

$\frac{x^{4} \cdot 4}{x^{4} (x^{7} y^{2})}$

Step 8.2.3

Rewrite the expression.

$\frac{4}{x^{7} y^{2}}$