Precalculus Examples

2 n^{- \frac{2}{3}} (n^{\frac{8}{3}} - 3 n^{\frac{5}{3}})

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

Step 1

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

2 \frac{1}{n^{\frac{2}{3}}} (n^{\frac{8}{3}} - 3 n^{\frac{5}{3}})

$2 \frac{1}{n^{\frac{2}{3}}} (n^{\frac{8}{3}} - 3 n^{\frac{5}{3}})$

Step 2

Combine fractions.

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Step 2.1

Combine

2

$2$ and

\frac{1}{n^{\frac{2}{3}}}

$\frac{1}{n^{\frac{2}{3}}}$ .

\frac{2}{n^{\frac{2}{3}}} (n^{\frac{8}{3}} - 3 n^{\frac{5}{3}})

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

Step 2.2

Multiply

\frac{2}{n^{\frac{2}{3}}}

$\frac{2}{n^{\frac{2}{3}}}$ by

n^{\frac{8}{3}} - 3 n^{\frac{5}{3}}

$n^{\frac{8}{3}} - 3 n^{\frac{5}{3}}$ .

\frac{2 (n^{\frac{8}{3}} - 3 n^{\frac{5}{3}})}{n^{\frac{2}{3}}}

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

\frac{2 (n^{\frac{8}{3}} - 3 n^{\frac{5}{3}})}{n^{\frac{2}{3}}}

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

Step 3

Simplify the numerator.

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Step 3.1

Factor

n^{\frac{5}{3}}

$n^{\frac{5}{3}}$ out of

n^{\frac{8}{3}} - 3 n^{\frac{5}{3}}

$n^{\frac{8}{3}} - 3 n^{\frac{5}{3}}$ .

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Step 3.1.1

Factor

n^{\frac{5}{3}}

$n^{\frac{5}{3}}$ out of

n^{\frac{8}{3}}

$n^{\frac{8}{3}}$ .

\frac{2 (n^{\frac{5}{3}} n^{\frac{3}{3}} - 3 n^{\frac{5}{3}})}{n^{\frac{2}{3}}}

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

Step 3.1.2

Factor

n^{\frac{5}{3}}

$n^{\frac{5}{3}}$ out of

- 3 n^{\frac{5}{3}}

$- 3 n^{\frac{5}{3}}$ .

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

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

Step 3.1.3

Factor

n^{\frac{5}{3}}

$n^{\frac{5}{3}}$ out of

n^{\frac{5}{3}} n^{\frac{3}{3}} + n^{\frac{5}{3}} \cdot - 3

$n^{\frac{5}{3}} n^{\frac{3}{3}} + n^{\frac{5}{3}} \cdot - 3$ .

\frac{2 (n^{\frac{5}{3}} (n^{\frac{3}{3}} - 3))}{n^{\frac{2}{3}}}

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

\frac{2 (n^{\frac{5}{3}} (n^{\frac{3}{3}} - 3))}{n^{\frac{2}{3}}}

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

Step 3.2

Divide

3

$3$ by

3

$3$ .

\frac{2 (n^{\frac{5}{3}} (n^{1} - 3))}{n^{\frac{2}{3}}}

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

Step 3.3

Simplify.

\frac{2 n^{\frac{5}{3}} (n - 3)}{n^{\frac{2}{3}}}

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

\frac{2 n^{\frac{5}{3}} (n - 3)}{n^{\frac{2}{3}}}

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

Step 4

Move

n^{\frac{2}{3}}

$n^{\frac{2}{3}}$ to the numerator using the negative exponent rule

\frac{1}{b^{n}} = b^{- n}

$\frac{1}{b^{n}} = b^{- n}$ .

2 n^{\frac{5}{3}} (n - 3) n^{- \frac{2}{3}}

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

Step 5

Multiply

n^{\frac{5}{3}}

$n^{\frac{5}{3}}$ by

n^{- \frac{2}{3}}

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

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Step 5.1

Move

n^{- \frac{2}{3}}

$n^{- \frac{2}{3}}$ .

2 (n^{- \frac{2}{3}} n^{\frac{5}{3}}) (n - 3)

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

Step 5.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 5.3

Combine the numerators over the common denominator.

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

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

Step 5.4

Add

- 2

$- 2$ and

5

$5$ .

2 n^{\frac{3}{3}} (n - 3)

$2 n^{\frac{3}{3}} (n - 3)$

Step 5.5

Divide

3

$3$ by

3

$3$ .

2 n^{1} (n - 3)

$2 n^{1} (n - 3)$

2 n^{1} (n - 3)

$2 n^{1} (n - 3)$

Step 6

Simplify

2 n^{1} (n - 3)

$2 n^{1} (n - 3)$ .

2 n (n - 3)

$2 n (n - 3)$

Step 7

Apply the distributive property.

2 n \cdot n + 2 n \cdot - 3

$2 n \cdot n + 2 n \cdot - 3$

Step 8

Multiply

n

$n$ by

n

$n$ by adding the exponents.

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Step 8.1

Move

n

$n$ .

2 (n \cdot n) + 2 n \cdot - 3

$2 (n \cdot n) + 2 n \cdot - 3$

Step 8.2

Multiply

n

$n$ by

n

$n$ .

2 n^{2} + 2 n \cdot - 3

$2 n^{2} + 2 n \cdot - 3$

2 n^{2} + 2 n \cdot - 3

$2 n^{2} + 2 n \cdot - 3$

Step 9

Multiply

- 3

$- 3$ by

2

$2$ .

2 n^{2} - 6 n

$2 n^{2} - 6 n$