Precalculus Examples

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

Step 1

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

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

Step 2

Multiply $x^{\frac{5}{2}}$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 2.4

Add $- 1$ and $5$ .

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

Step 2.5

Divide $4$ by $2$ .

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

Step 3

Apply the product rule to $2 x^{\frac{5}{4}}$ .

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

Step 4

Rewrite using the commutative property of multiplication.

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

Step 5

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

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

Step 6

Multiply the exponents in ${(x^{\frac{5}{4}})}^{2}$ .

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Step 6.1

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

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

Step 6.2

Cancel the common factor of $2$ .

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Step 6.2.1

Factor $2$ out of $4$ .

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

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

Rewrite the expression.

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

Step 7

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

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Step 7.1

Move $x^{\frac{5}{2}}$ .

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

Step 7.2

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

$4 x^{\frac{5}{2} + 2}$

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

Combine the numerators over the common denominator.

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

Step 7.6

Simplify the numerator.

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Step 7.6.1

Multiply $2$ by $2$ .

$4 x^{\frac{5 + 4}{2}}$

Step 7.6.2

Add $5$ and $4$ .

$4 x^{\frac{9}{2}}$