Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Cancel the common factor of $2$ .

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Step 1.2.2.1

Cancel the common factor.

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

Step 1.2.2.2

Rewrite the expression.

$\frac{2^{\frac{3}{2}} x^{3}}{2^{\frac{1}{2}} x^{4}}$

Step 2

Move $2^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{2^{\frac{3}{2}} x^{3} \cdot 2^{- \frac{1}{2}}}{x^{4}}$

Step 3

Simplify the numerator.

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Step 3.1

Multiply $2^{\frac{3}{2}}$ by $2^{- \frac{1}{2}}$ by adding the exponents.

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Step 3.1.1

Move $2^{- \frac{1}{2}}$ .

$\frac{2^{- \frac{1}{2}} \cdot 2^{\frac{3}{2}} x^{3}}{x^{4}}$

Step 3.1.2

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

$\frac{2^{- \frac{1}{2} + \frac{3}{2}} x^{3}}{x^{4}}$

Step 3.1.3

Combine the numerators over the common denominator.

$\frac{2^{\frac{- 1 + 3}{2}} x^{3}}{x^{4}}$

Step 3.1.4

Add $- 1$ and $3$ .

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

Step 3.1.5

Divide $2$ by $2$ .

$\frac{2^{1} x^{3}}{x^{4}}$

Step 3.2

Simplify $2^{1} x^{3}$ .

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

Step 4

Cancel the common factor of $x^{3}$ and $x^{4}$ .

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Step 4.1

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

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

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $x^{3}$ out of $x^{4}$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

$\frac{2}{x}$