Finite Math Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Apply the product rule to $x y$ .

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

Step 1.2

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

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

Step 1.3

Multiply the exponents in ${(x^{2})}^{\frac{1}{2}}$ .

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Step 1.3.1

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

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

Step 1.3.2

Cancel the common factor of $2$ .

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Step 1.3.2.1

Cancel the common factor.

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

Step 1.3.2.2

Rewrite the expression.

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

Step 1.4

Simplify.

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

Step 1.5

Multiply the exponents in ${(y^{2})}^{\frac{1}{2}}$ .

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Step 1.5.1

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

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

Step 1.5.2

Cancel the common factor of $2$ .

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Step 1.5.2.1

Cancel the common factor.

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

Step 1.5.2.2

Rewrite the expression.

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

Step 1.6

Simplify.

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

Step 1.7

Combine exponents.

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Step 1.7.1

Raise $x$ to the power of $1$ .

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

Step 1.7.2

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

$\frac{x^{1 + \frac{1}{4}} y^{\frac{1}{4}} y}{{(x^{2} y)}^{\frac{3}{4}}}$

Step 1.7.3

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

$\frac{x^{\frac{4}{4} + \frac{1}{4}} y^{\frac{1}{4}} y}{{(x^{2} y)}^{\frac{3}{4}}}$

Step 1.7.4

Combine the numerators over the common denominator.

$\frac{x^{\frac{4 + 1}{4}} y^{\frac{1}{4}} y}{{(x^{2} y)}^{\frac{3}{4}}}$

Step 1.7.5

Add $4$ and $1$ .

$\frac{x^{\frac{5}{4}} y^{\frac{1}{4}} y}{{(x^{2} y)}^{\frac{3}{4}}}$

Step 1.7.6

Raise $y$ to the power of $1$ .

$\frac{x^{\frac{5}{4}} (y^{1} y^{\frac{1}{4}})}{{(x^{2} y)}^{\frac{3}{4}}}$

Step 1.7.7

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

$\frac{x^{\frac{5}{4}} y^{1 + \frac{1}{4}}}{{(x^{2} y)}^{\frac{3}{4}}}$

Step 1.7.8

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

$\frac{x^{\frac{5}{4}} y^{\frac{4}{4} + \frac{1}{4}}}{{(x^{2} y)}^{\frac{3}{4}}}$

Step 1.7.9

Combine the numerators over the common denominator.

$\frac{x^{\frac{5}{4}} y^{\frac{4 + 1}{4}}}{{(x^{2} y)}^{\frac{3}{4}}}$

Step 1.7.10

Add $4$ and $1$ .

$\frac{x^{\frac{5}{4}} y^{\frac{5}{4}}}{{(x^{2} y)}^{\frac{3}{4}}}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{x^{\frac{5}{4}} y^{\frac{5}{4}}}{{(x^{2})}^{\frac{3}{4}} y^{\frac{3}{4}}}$

Step 2.2

Multiply the exponents in ${(x^{2})}^{\frac{3}{4}}$ .

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Step 2.2.1

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

$\frac{x^{\frac{5}{4}} y^{\frac{5}{4}}}{x^{2 (\frac{3}{4})} y^{\frac{3}{4}}}$

Step 2.2.2

Cancel the common factor of $2$ .

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Step 2.2.2.1

Factor $2$ out of $4$ .

$\frac{x^{\frac{5}{4}} y^{\frac{5}{4}}}{x^{2 \frac{3}{2 (2)}} y^{\frac{3}{4}}}$

Step 2.2.2.2

Cancel the common factor.

$\frac{x^{\frac{5}{4}} y^{\frac{5}{4}}}{x^{2 \frac{3}{2 \cdot 2}} y^{\frac{3}{4}}}$

Step 2.2.2.3

Rewrite the expression.

$\frac{x^{\frac{5}{4}} y^{\frac{5}{4}}}{x^{\frac{3}{2}} y^{\frac{3}{4}}}$

Step 3

Move $x^{\frac{5}{4}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{y^{\frac{5}{4}}}{x^{\frac{3}{2}} y^{\frac{3}{4}} x^{- \frac{5}{4}}}$

Step 4

Multiply $x^{\frac{3}{2}}$ by $x^{- \frac{5}{4}}$ by adding the exponents.

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Step 4.1

Move $x^{- \frac{5}{4}}$ .

$\frac{y^{\frac{5}{4}}}{x^{- \frac{5}{4}} x^{\frac{3}{2}} y^{\frac{3}{4}}}$

Step 4.2

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

$\frac{y^{\frac{5}{4}}}{x^{- \frac{5}{4} + \frac{3}{2}} y^{\frac{3}{4}}}$

Step 4.3

To write $\frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{y^{\frac{5}{4}}}{x^{- \frac{5}{4} + \frac{3}{2} \cdot \frac{2}{2}} y^{\frac{3}{4}}}$

Step 4.4

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.4.1

Multiply $\frac{3}{2}$ by $\frac{2}{2}$ .

$\frac{y^{\frac{5}{4}}}{x^{- \frac{5}{4} + \frac{3 \cdot 2}{2 \cdot 2}} y^{\frac{3}{4}}}$

Step 4.4.2

Multiply $2$ by $2$ .

$\frac{y^{\frac{5}{4}}}{x^{- \frac{5}{4} + \frac{3 \cdot 2}{4}} y^{\frac{3}{4}}}$

Step 4.5

Combine the numerators over the common denominator.

$\frac{y^{\frac{5}{4}}}{x^{\frac{- 5 + 3 \cdot 2}{4}} y^{\frac{3}{4}}}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $3$ by $2$ .

$\frac{y^{\frac{5}{4}}}{x^{\frac{- 5 + 6}{4}} y^{\frac{3}{4}}}$

Step 4.6.2

Add $- 5$ and $6$ .

$\frac{y^{\frac{5}{4}}}{x^{\frac{1}{4}} y^{\frac{3}{4}}}$

Step 5

Move $y^{\frac{3}{4}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{y^{\frac{5}{4}} y^{- \frac{3}{4}}}{x^{\frac{1}{4}}}$

Step 6

Multiply $y^{\frac{5}{4}}$ by $y^{- \frac{3}{4}}$ by adding the exponents.

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Step 6.1

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

$\frac{y^{\frac{5}{4} - \frac{3}{4}}}{x^{\frac{1}{4}}}$

Step 6.2

Combine the numerators over the common denominator.

$\frac{y^{\frac{5 - 3}{4}}}{x^{\frac{1}{4}}}$

Step 6.3

Subtract $3$ from $5$ .

$\frac{y^{\frac{2}{4}}}{x^{\frac{1}{4}}}$

Step 6.4

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

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Step 6.4.1

Factor $2$ out of $2$ .

$\frac{y^{\frac{2 (1)}{4}}}{x^{\frac{1}{4}}}$

Step 6.4.2

Cancel the common factors.

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Step 6.4.2.1

Factor $2$ out of $4$ .

$\frac{y^{\frac{2 \cdot 1}{2 \cdot 2}}}{x^{\frac{1}{4}}}$

Step 6.4.2.2

Cancel the common factor.

$\frac{y^{\frac{2 \cdot 1}{2 \cdot 2}}}{x^{\frac{1}{4}}}$

Step 6.4.2.3

Rewrite the expression.

$\frac{y^{\frac{1}{2}}}{x^{\frac{1}{4}}}$