Basic Math Examples

$(- \frac{4}{9}) \cdot (3^{- \frac{5}{3}}) + (\frac{2}{9}) (3^{- \frac{4}{3}})$

Step 1

Simplify each term.

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Step 1.1

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

$- \frac{4}{9} \cdot \frac{1}{3^{\frac{5}{3}}} + (\frac{2}{9}) (3^{- \frac{4}{3}})$

Step 1.2

Multiply $- \frac{4}{9} \cdot \frac{1}{3^{\frac{5}{3}}}$ .

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Step 1.2.1

Multiply $\frac{1}{3^{\frac{5}{3}}}$ by $\frac{4}{9}$ .

$- \frac{4}{3^{\frac{5}{3}} \cdot 9} + (\frac{2}{9}) (3^{- \frac{4}{3}})$

Step 1.2.2

Rewrite $9$ as $3^{2}$ .

$- \frac{4}{3^{\frac{5}{3}} \cdot 3^{2}} + (\frac{2}{9}) (3^{- \frac{4}{3}})$

Step 1.2.3

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

$- \frac{4}{3^{\frac{5}{3} + 2}} + (\frac{2}{9}) (3^{- \frac{4}{3}})$

Step 1.2.4

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

$- \frac{4}{3^{\frac{5}{3} + 2 \cdot \frac{3}{3}}} + (\frac{2}{9}) (3^{- \frac{4}{3}})$

Step 1.2.5

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

$- \frac{4}{3^{\frac{5}{3} + \frac{2 \cdot 3}{3}}} + (\frac{2}{9}) (3^{- \frac{4}{3}})$

Step 1.2.6

Combine the numerators over the common denominator.

$- \frac{4}{3^{\frac{5 + 2 \cdot 3}{3}}} + (\frac{2}{9}) (3^{- \frac{4}{3}})$

Step 1.2.7

Simplify the numerator.

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Step 1.2.7.1

Multiply $2$ by $3$ .

$- \frac{4}{3^{\frac{5 + 6}{3}}} + (\frac{2}{9}) (3^{- \frac{4}{3}})$

Step 1.2.7.2

Add $5$ and $6$ .

$- \frac{4}{3^{\frac{11}{3}}} + (\frac{2}{9}) (3^{- \frac{4}{3}})$

Step 1.3

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

$- \frac{4}{3^{\frac{11}{3}}} + \frac{2}{9} \cdot \frac{1}{3^{\frac{4}{3}}}$

Step 1.4

Combine.

$- \frac{4}{3^{\frac{11}{3}}} + \frac{2 \cdot 1}{9 \cdot 3^{\frac{4}{3}}}$

Step 1.5

Multiply $2$ by $1$ .

$- \frac{4}{3^{\frac{11}{3}}} + \frac{2}{9 \cdot 3^{\frac{4}{3}}}$

Step 1.6

Simplify the denominator.

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Step 1.6.1

Rewrite $9$ as $3^{2}$ .

$- \frac{4}{3^{\frac{11}{3}}} + \frac{2}{3^{2} \cdot 3^{\frac{4}{3}}}$

Step 1.6.2

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

$- \frac{4}{3^{\frac{11}{3}}} + \frac{2}{3^{2 + \frac{4}{3}}}$

Step 1.6.3

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

$- \frac{4}{3^{\frac{11}{3}}} + \frac{2}{3^{2 \cdot \frac{3}{3} + \frac{4}{3}}}$

Step 1.6.4

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

$- \frac{4}{3^{\frac{11}{3}}} + \frac{2}{3^{\frac{2 \cdot 3}{3} + \frac{4}{3}}}$

Step 1.6.5

Combine the numerators over the common denominator.

$- \frac{4}{3^{\frac{11}{3}}} + \frac{2}{3^{\frac{2 \cdot 3 + 4}{3}}}$

Step 1.6.6

Simplify the numerator.

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Step 1.6.6.1

Multiply $2$ by $3$ .

$- \frac{4}{3^{\frac{11}{3}}} + \frac{2}{3^{\frac{6 + 4}{3}}}$

Step 1.6.6.2

Add $6$ and $4$ .

$- \frac{4}{3^{\frac{11}{3}}} + \frac{2}{3^{\frac{10}{3}}}$

Step 2

To write $\frac{2}{3^{\frac{10}{3}}}$ as a fraction with a common denominator, multiply by $\frac{3^{\frac{1}{3}}}{3^{\frac{1}{3}}}$ .

$- \frac{4}{3^{\frac{11}{3}}} + \frac{2}{3^{\frac{10}{3}}} \cdot \frac{3^{\frac{1}{3}}}{3^{\frac{1}{3}}}$

Step 3

Write each expression with a common denominator of $3^{\frac{11}{3}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{2}{3^{\frac{10}{3}}}$ by $\frac{3^{\frac{1}{3}}}{3^{\frac{1}{3}}}$ .

$- \frac{4}{3^{\frac{11}{3}}} + \frac{2 \cdot 3^{\frac{1}{3}}}{3^{\frac{10}{3}} \cdot 3^{\frac{1}{3}}}$

Step 3.2

Multiply $3^{\frac{10}{3}}$ by $3^{\frac{1}{3}}$ by adding the exponents.

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Step 3.2.1

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

$- \frac{4}{3^{\frac{11}{3}}} + \frac{2 \cdot 3^{\frac{1}{3}}}{3^{\frac{10}{3} + \frac{1}{3}}}$

Step 3.2.2

Combine the numerators over the common denominator.

$- \frac{4}{3^{\frac{11}{3}}} + \frac{2 \cdot 3^{\frac{1}{3}}}{3^{\frac{10 + 1}{3}}}$

Step 3.2.3

Add $10$ and $1$ .

$- \frac{4}{3^{\frac{11}{3}}} + \frac{2 \cdot 3^{\frac{1}{3}}}{3^{\frac{11}{3}}}$

Step 4

Combine the numerators over the common denominator.

$\frac{- 4 + 2 \cdot 3^{\frac{1}{3}}}{3^{\frac{11}{3}}}$

Step 5

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{- 1 (4) + 2 \cdot 3^{\frac{1}{3}}}{3^{\frac{11}{3}}}$

Step 6

Factor $- 1$ out of $2 \cdot 3^{\frac{1}{3}}$ .

$\frac{- 1 (4) - (- 2 \cdot 3^{\frac{1}{3}})}{3^{\frac{11}{3}}}$

Step 7

Factor $- 1$ out of $- 1 (4) - (- 2 \cdot 3^{\frac{1}{3}})$ .

$\frac{- 1 (4 - 2 \cdot 3^{\frac{1}{3}})}{3^{\frac{11}{3}}}$

Step 8

Move the negative in front of the fraction.

$- \frac{4 - 2 \cdot 3^{\frac{1}{3}}}{3^{\frac{11}{3}}}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{4 - 2 \cdot 3^{\frac{1}{3}}}{3^{\frac{11}{3}}}$

Decimal Form:

$- 0.01986210 \dots$