Precalculus Examples

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

Step 1

Move $x^{- \frac{2}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 2

Simplify the numerator.

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Step 2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.2

To write $3 x^{\frac{1}{3}}$ as a fraction with a common denominator, multiply by $\frac{x^{\frac{2}{3}}}{x^{\frac{2}{3}}}$ .

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 2.4

Rewrite $\frac{3 x^{\frac{1}{3}} x^{\frac{2}{3}} - 1}{x^{\frac{2}{3}}}$ in a factored form.

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Step 2.4.1

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

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Step 2.4.1.1

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

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

Step 2.4.1.2

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

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

Step 2.4.1.3

Combine the numerators over the common denominator.

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

Step 2.4.1.4

Add $2$ and $1$ .

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

Step 2.4.1.5

Divide $3$ by $3$ .

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

Step 2.4.2

Simplify $3 x^{1}$ .

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

Step 3

Combine $\frac{3 x - 1}{x^{\frac{2}{3}}}$ and $x^{\frac{2}{3}}$ .

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

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Reduce the expression $\frac{(3 x - 1) x^{\frac{2}{3}}}{x^{\frac{2}{3}}}$ by cancelling the common factors.

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Step 4.1.1

Cancel the common factor.

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

Step 4.1.2

Rewrite the expression.

$\frac{\frac{3 x - 1}{1}}{3}$

Step 4.2

Divide $3 x - 1$ by $1$ .

$\frac{3 x - 1}{3}$