Algebra Examples

$\frac{3 x^{\frac{7}{2}} y^{- 4}}{\sqrt[4]{16 x^{2} y^{4}}}$

Step 1

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

$\frac{3 x^{\frac{7}{2}}}{\sqrt[4]{16 x^{2} y^{4}} y^{4}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $16 x^{2} y^{4}$ as ${(2 y)}^{4} x^{2}$ .

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Step 2.1.1

Rewrite $16$ as $2^{4}$ .

$\frac{3 x^{\frac{7}{2}}}{\sqrt[4]{2^{4} x^{2} y^{4}} y^{4}}$

Step 2.1.2

Move $x^{2}$ .

$\frac{3 x^{\frac{7}{2}}}{\sqrt[4]{2^{4} y^{4} x^{2}} y^{4}}$

Step 2.1.3

Rewrite $2^{4} y^{4}$ as ${(2 y)}^{4}$ .

$\frac{3 x^{\frac{7}{2}}}{\sqrt[4]{{(2 y)}^{4} x^{2}} y^{4}}$

Step 2.2

Pull terms out from under the radical.

$\frac{3 x^{\frac{7}{2}}}{2 y \sqrt[4]{x^{2}} y^{4}}$

Step 2.3

Rewrite $\sqrt[4]{x^{2}}$ as $\sqrt{\sqrt{x^{2}}}$ .

$\frac{3 x^{\frac{7}{2}}}{2 y \sqrt{\sqrt{x^{2}}} y^{4}}$

Step 2.4

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

$\frac{3 x^{\frac{7}{2}}}{2 y \sqrt{x} y^{4}}$

Step 2.5

Multiply $y$ by $y^{4}$ by adding the exponents.

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Step 2.5.1

Move $y^{4}$ .

$\frac{3 x^{\frac{7}{2}}}{2 (y^{4} y) \sqrt{x}}$

Step 2.5.2

Multiply $y^{4}$ by $y$ .

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Step 2.5.2.1

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

$\frac{3 x^{\frac{7}{2}}}{2 (y^{4} y^{1}) \sqrt{x}}$

Step 2.5.2.2

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

$\frac{3 x^{\frac{7}{2}}}{2 y^{4 + 1} \sqrt{x}}$

Step 2.5.3

Add $4$ and $1$ .

$\frac{3 x^{\frac{7}{2}}}{2 y^{5} \sqrt{x}}$

Step 3

Multiply $\frac{3 x^{\frac{7}{2}}}{2 y^{5} \sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{3 x^{\frac{7}{2}}}{2 y^{5} \sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{3 x^{\frac{7}{2}}}{2 y^{5} \sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{3 x^{\frac{7}{2}} \sqrt{x}}{2 y^{5} \sqrt{x} \sqrt{x}}$

Step 4.2

Move $\sqrt{x}$ .

$\frac{3 x^{\frac{7}{2}} \sqrt{x}}{2 y^{5} (\sqrt{x} \sqrt{x})}$

Step 4.3

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

$\frac{3 x^{\frac{7}{2}} \sqrt{x}}{2 y^{5} ({\sqrt{x}}^{1} \sqrt{x})}$

Step 4.4

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

$\frac{3 x^{\frac{7}{2}} \sqrt{x}}{2 y^{5} ({\sqrt{x}}^{1} {\sqrt{x}}^{1})}$

Step 4.5

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

$\frac{3 x^{\frac{7}{2}} \sqrt{x}}{2 y^{5} {\sqrt{x}}^{1 + 1}}$

Step 4.6

Add $1$ and $1$ .

$\frac{3 x^{\frac{7}{2}} \sqrt{x}}{2 y^{5} {\sqrt{x}}^{2}}$

Step 4.7

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

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Step 4.7.1

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

$\frac{3 x^{\frac{7}{2}} \sqrt{x}}{2 y^{5} {(x^{\frac{1}{2}})}^{2}}$

Step 4.7.2

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

$\frac{3 x^{\frac{7}{2}} \sqrt{x}}{2 y^{5} x^{\frac{1}{2} \cdot 2}}$

Step 4.7.3

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

$\frac{3 x^{\frac{7}{2}} \sqrt{x}}{2 y^{5} x^{\frac{2}{2}}}$

Step 4.7.4

Cancel the common factor of $2$ .

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Step 4.7.4.1

Cancel the common factor.

$\frac{3 x^{\frac{7}{2}} \sqrt{x}}{2 y^{5} x^{\frac{2}{2}}}$

Step 4.7.4.2

Rewrite the expression.

$\frac{3 x^{\frac{7}{2}} \sqrt{x}}{2 y^{5} x^{1}}$

Step 4.7.5

Simplify.

$\frac{3 x^{\frac{7}{2}} \sqrt{x}}{2 y^{5} x}$

Step 5

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

$\frac{3 x^{\frac{7}{2}} \sqrt{x} x^{- 1}}{2 y^{5}}$

Step 6

Multiply $x^{\frac{7}{2}}$ by $x^{- 1}$ by adding the exponents.

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Step 6.1

Move $x^{- 1}$ .

$\frac{3 (x^{- 1} x^{\frac{7}{2}}) \sqrt{x}}{2 y^{5}}$

Step 6.2

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

$\frac{3 x^{- 1 + \frac{7}{2}} \sqrt{x}}{2 y^{5}}$

Step 6.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 x^{- 1 \cdot \frac{2}{2} + \frac{7}{2}} \sqrt{x}}{2 y^{5}}$

Step 6.4

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

$\frac{3 x^{\frac{- 1 \cdot 2}{2} + \frac{7}{2}} \sqrt{x}}{2 y^{5}}$

Step 6.5

Combine the numerators over the common denominator.

$\frac{3 x^{\frac{- 1 \cdot 2 + 7}{2}} \sqrt{x}}{2 y^{5}}$

Step 6.6

Simplify the numerator.

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Step 6.6.1

Multiply $- 1$ by $2$ .

$\frac{3 x^{\frac{- 2 + 7}{2}} \sqrt{x}}{2 y^{5}}$

Step 6.6.2

Add $- 2$ and $7$ .

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

Step 7

Simplify the numerator.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

Combine the numerators over the common denominator.

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

Step 7.4

Add $1$ and $5$ .

$\frac{3 x^{\frac{6}{2}}}{2 y^{5}}$

Step 7.5

Cancel the common factor of $6$ and $2$ .

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Step 7.5.1

Factor $2$ out of $6$ .

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

Step 7.5.2

Cancel the common factors.

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Step 7.5.2.1

Factor $2$ out of $2$ .

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

Step 7.5.2.2

Cancel the common factor.

$\frac{3 x^{\frac{2 \cdot 3}{2 \cdot 1}}}{2 y^{5}}$

Step 7.5.2.3

Rewrite the expression.

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

Step 7.5.2.4

Divide $3$ by $1$ .

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