Precalculus Examples

$x^{\frac{3}{2}} (x + x^{\frac{5}{2}} - x^{2})$

Step 1

Apply the distributive property.

$x^{\frac{3}{2}} x + x^{\frac{3}{2}} x^{\frac{5}{2}} + x^{\frac{3}{2}} (- x^{2})$

Step 2

Simplify.

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Step 2.1

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

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Step 2.1.1

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

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Step 2.1.1.1

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

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

Step 2.1.1.2

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

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

Step 2.1.2

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

$x^{\frac{3}{2} + \frac{2}{2}} + x^{\frac{3}{2}} x^{\frac{5}{2}} + x^{\frac{3}{2}} (- x^{2})$

Step 2.1.3

Combine the numerators over the common denominator.

$x^{\frac{3 + 2}{2}} + x^{\frac{3}{2}} x^{\frac{5}{2}} + x^{\frac{3}{2}} (- x^{2})$

Step 2.1.4

Add $3$ and $2$ .

$x^{\frac{5}{2}} + x^{\frac{3}{2}} x^{\frac{5}{2}} + x^{\frac{3}{2}} (- x^{2})$

Step 2.2

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

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Step 2.2.1

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

$x^{\frac{5}{2}} + x^{\frac{3}{2} + \frac{5}{2}} + x^{\frac{3}{2}} (- x^{2})$

Step 2.2.2

Combine the numerators over the common denominator.

$x^{\frac{5}{2}} + x^{\frac{3 + 5}{2}} + x^{\frac{3}{2}} (- x^{2})$

Step 2.2.3

Add $3$ and $5$ .

$x^{\frac{5}{2}} + x^{\frac{8}{2}} + x^{\frac{3}{2}} (- x^{2})$

Step 2.2.4

Divide $8$ by $2$ .

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

Step 2.3

Rewrite using the commutative property of multiplication.

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

Step 3

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

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Step 3.1

Move $x^{2}$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $2$ by $2$ .

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

Step 3.6.2

Add $4$ and $3$ .

$x^{\frac{5}{2}} + x^{4} - x^{\frac{7}{2}}$

Step 4

Reorder $x^{\frac{5}{2}}$ and $x^{4}$ .

$x^{4} + x^{\frac{5}{2}} - x^{\frac{7}{2}}$