Precalculus Examples

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

Step 1

Apply the distributive property.

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

Step 2

Simplify.

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Step 2.1

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

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Step 2.1.1

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

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Step 2.1.1.1

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

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

Step 2.1.1.2

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

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

Step 2.1.2

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

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

Step 2.1.3

Combine the numerators over the common denominator.

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

Step 2.1.4

Add $2$ and $3$ .

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

Step 2.2

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

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

Step 2.2.2

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

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

Step 2.2.3

To write $\frac{5}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 2.2.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.4.1

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

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

Step 2.2.4.2

Multiply $3$ by $2$ .

$x^{\frac{5}{3}} + x^{\frac{2 \cdot 2}{6} + \frac{5}{2} \cdot \frac{3}{3}} + x^{\frac{2}{3}} (- x^{2})$

Step 2.2.4.3

Multiply $\frac{5}{2}$ by $\frac{3}{3}$ .

$x^{\frac{5}{3}} + x^{\frac{2 \cdot 2}{6} + \frac{5 \cdot 3}{2 \cdot 3}} + x^{\frac{2}{3}} (- x^{2})$

Step 2.2.4.4

Multiply $2$ by $3$ .

$x^{\frac{5}{3}} + x^{\frac{2 \cdot 2}{6} + \frac{5 \cdot 3}{6}} + x^{\frac{2}{3}} (- x^{2})$

Step 2.2.5

Combine the numerators over the common denominator.

$x^{\frac{5}{3}} + x^{\frac{2 \cdot 2 + 5 \cdot 3}{6}} + x^{\frac{2}{3}} (- x^{2})$

Step 2.2.6

Simplify the numerator.

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Step 2.2.6.1

Multiply $2$ by $2$ .

$x^{\frac{5}{3}} + x^{\frac{4 + 5 \cdot 3}{6}} + x^{\frac{2}{3}} (- x^{2})$

Step 2.2.6.2

Multiply $5$ by $3$ .

$x^{\frac{5}{3}} + x^{\frac{4 + 15}{6}} + x^{\frac{2}{3}} (- x^{2})$

Step 2.2.6.3

Add $4$ and $15$ .

$x^{\frac{5}{3}} + x^{\frac{19}{6}} + x^{\frac{2}{3}} (- x^{2})$

Step 2.3

Rewrite using the commutative property of multiplication.

$x^{\frac{5}{3}} + x^{\frac{19}{6}} - x^{\frac{2}{3}} x^{2}$

Step 3

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

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Step 3.1

Move $x^{2}$ .

$x^{\frac{5}{3}} + x^{\frac{19}{6}} - (x^{2} x^{\frac{2}{3}})$

Step 3.2

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

$x^{\frac{5}{3}} + x^{\frac{19}{6}} - x^{2 + \frac{2}{3}}$

Step 3.3

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

$x^{\frac{5}{3}} + x^{\frac{19}{6}} - x^{2 \cdot \frac{3}{3} + \frac{2}{3}}$

Step 3.4

Combine $2$ and $\frac{3}{3}$ .

$x^{\frac{5}{3}} + x^{\frac{19}{6}} - x^{\frac{2 \cdot 3}{3} + \frac{2}{3}}$

Step 3.5

Combine the numerators over the common denominator.

$x^{\frac{5}{3}} + x^{\frac{19}{6}} - x^{\frac{2 \cdot 3 + 2}{3}}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $2$ by $3$ .

$x^{\frac{5}{3}} + x^{\frac{19}{6}} - x^{\frac{6 + 2}{3}}$

Step 3.6.2

Add $6$ and $2$ .

$x^{\frac{5}{3}} + x^{\frac{19}{6}} - x^{\frac{8}{3}}$