Precalculus Examples

$a = \frac{7 (1 - {(- \frac{2}{5})}^{3})}{1 - (- \frac{2}{5})}$

Step 1

Simplify the numerator.

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Step 1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.1.1

Apply the product rule to $- \frac{2}{5}$ .

$a = \frac{7 (1 - ({(- 1)}^{3} {(\frac{2}{5})}^{3}))}{1 + \frac{2}{5}}$

Step 1.1.2

Apply the product rule to $\frac{2}{5}$ .

$a = \frac{7 (1 - ({(- 1)}^{3} \frac{2^{3}}{5^{3}}))}{1 + \frac{2}{5}}$

Step 1.2

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

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Step 1.2.1

Move ${(- 1)}^{3}$ .

$a = \frac{7 (1 + {(- 1)}^{3} \cdot - 1 \frac{2^{3}}{5^{3}})}{1 + \frac{2}{5}}$

Step 1.2.2

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

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Step 1.2.2.1

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

$a = \frac{7 (1 + {(- 1)}^{3} \cdot {(- 1)}^{1} \frac{2^{3}}{5^{3}})}{1 + \frac{2}{5}}$

Step 1.2.2.2

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

$a = \frac{7 (1 + {(- 1)}^{3 + 1} \frac{2^{3}}{5^{3}})}{1 + \frac{2}{5}}$

Step 1.2.3

Add $3$ and $1$ .

$a = \frac{7 (1 + {(- 1)}^{4} \frac{2^{3}}{5^{3}})}{1 + \frac{2}{5}}$

Step 1.3

Raise $- 1$ to the power of $4$ .

$a = \frac{7 (1 + 1 \frac{2^{3}}{5^{3}})}{1 + \frac{2}{5}}$

Step 1.4

Multiply $\frac{2^{3}}{5^{3}}$ by $1$ .

$a = \frac{7 (1 + \frac{2^{3}}{5^{3}})}{1 + \frac{2}{5}}$

Step 1.5

Raise $2$ to the power of $3$ .

$a = \frac{7 (1 + \frac{8}{5^{3}})}{1 + \frac{2}{5}}$

Step 1.6

Raise $5$ to the power of $3$ .

$a = \frac{7 (1 + \frac{8}{125})}{1 + \frac{2}{5}}$

Step 1.7

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

$a = \frac{7 (\frac{125}{125} + \frac{8}{125})}{1 + \frac{2}{5}}$

Step 1.8

Combine the numerators over the common denominator.

$a = \frac{7 \frac{125 + 8}{125}}{1 + \frac{2}{5}}$

Step 1.9

Add $125$ and $8$ .

$a = \frac{7 (\frac{133}{125})}{1 + \frac{2}{5}}$

Step 2

Simplify the denominator.

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Step 2.1

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

$a = \frac{7 (\frac{133}{125})}{\frac{5}{5} + \frac{2}{5}}$

Step 2.2

Combine the numerators over the common denominator.

$a = \frac{7 (\frac{133}{125})}{\frac{5 + 2}{5}}$

Step 2.3

Add $5$ and $2$ .

$a = \frac{7 (\frac{133}{125})}{\frac{7}{5}}$

Step 3

Combine fractions.

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Step 3.1

Combine $7$ and $\frac{133}{125}$ .

$a = \frac{\frac{7 \cdot 133}{125}}{\frac{7}{5}}$

Step 3.2

Multiply $7$ by $133$ .

$a = \frac{\frac{931}{125}}{\frac{7}{5}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$a = \frac{931}{125} \cdot \frac{5}{7}$

Step 5

Cancel the common factor of $7$ .

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Step 5.1

Factor $7$ out of $931$ .

$a = \frac{7 (133)}{125} \cdot \frac{5}{7}$

Step 5.2

Cancel the common factor.

$a = \frac{7 \cdot 133}{125} \cdot \frac{5}{7}$

Step 5.3

Rewrite the expression.

$a = \frac{133}{125} \cdot 5$

Step 6

Cancel the common factor of $5$ .

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Step 6.1

Factor $5$ out of $125$ .

$a = \frac{133}{5 (25)} \cdot 5$

Step 6.2

Cancel the common factor.

$a = \frac{133}{5 \cdot 25} \cdot 5$

Step 6.3

Rewrite the expression.

$a = \frac{133}{25}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$a = \frac{133}{25}$

Decimal Form:

$a = 5.32$

Mixed Number Form:

$a = 5 \frac{8}{25}$