Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 1.3

Multiply $2$ by $1$ .

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

Step 1.4

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

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

Step 1.5

Multiply $\frac{1}{{(- 2)}^{\frac{1}{3}}}$ by $\frac{2}{3}$ .

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

Step 1.6

Move $3$ to the left of ${(- 2)}^{\frac{1}{3}}$ .

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

Step 2

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

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

Step 3

Write each expression with a common denominator of $3 {(- 2)}^{\frac{2}{3}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{2}{3 {(- 2)}^{\frac{1}{3}}}$ by $\frac{{(- 2)}^{\frac{1}{3}}}{{(- 2)}^{\frac{1}{3}}}$ .

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

Step 3.2

Multiply ${(- 2)}^{\frac{1}{3}}$ by ${(- 2)}^{\frac{1}{3}}$ by adding the exponents.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

Combine the numerators over the common denominator.

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

Step 3.2.4

Add $1$ and $1$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Multiply $- 2$ by ${(- 2)}^{\frac{1}{3}}$ by adding the exponents.

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Step 5.1

Multiply $- 2$ by ${(- 2)}^{\frac{1}{3}}$ .

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Step 5.1.1

Raise $- 2$ to the power of $1$ .

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

Step 5.1.2

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

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

Step 5.2

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

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

Step 5.3

Combine the numerators over the common denominator.

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

Step 5.4

Add $3$ and $1$ .

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.94910736 \dots$