Precalculus Examples

$\sum_{k = 1}^{4} 72 {(\frac{1}{3})}^{k}$

Step 1

Expand the series for each value of $k$ .

$72 {(\frac{1}{3})}^{1} + 72 {(\frac{1}{3})}^{2} + 72 {(\frac{1}{3})}^{3} + 72 {(\frac{1}{3})}^{4}$

Step 2

Simplify.

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Step 2.1

Simplify each term.

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Step 2.1.1

Simplify.

$72 (\frac{1}{3}) + 72 {(\frac{1}{3})}^{2} + 72 {(\frac{1}{3})}^{3} + 72 {(\frac{1}{3})}^{4}$

Step 2.1.2

Cancel the common factor of $3$ .

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Step 2.1.2.1

Factor $3$ out of $72$ .

$3 (24) \frac{1}{3} + 72 {(\frac{1}{3})}^{2} + 72 {(\frac{1}{3})}^{3} + 72 {(\frac{1}{3})}^{4}$

Step 2.1.2.2

Cancel the common factor.

$3 \cdot 24 \frac{1}{3} + 72 {(\frac{1}{3})}^{2} + 72 {(\frac{1}{3})}^{3} + 72 {(\frac{1}{3})}^{4}$

Step 2.1.2.3

Rewrite the expression.

$24 + 72 {(\frac{1}{3})}^{2} + 72 {(\frac{1}{3})}^{3} + 72 {(\frac{1}{3})}^{4}$

Step 2.1.3

Apply the product rule to $\frac{1}{3}$ .

$24 + 72 \frac{1^{2}}{3^{2}} + 72 {(\frac{1}{3})}^{3} + 72 {(\frac{1}{3})}^{4}$

Step 2.1.4

One to any power is one.

$24 + 72 \frac{1}{3^{2}} + 72 {(\frac{1}{3})}^{3} + 72 {(\frac{1}{3})}^{4}$

Step 2.1.5

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

$24 + 72 (\frac{1}{9}) + 72 {(\frac{1}{3})}^{3} + 72 {(\frac{1}{3})}^{4}$

Step 2.1.6

Cancel the common factor of $9$ .

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Step 2.1.6.1

Factor $9$ out of $72$ .

$24 + 9 (8) \frac{1}{9} + 72 {(\frac{1}{3})}^{3} + 72 {(\frac{1}{3})}^{4}$

Step 2.1.6.2

Cancel the common factor.

$24 + 9 \cdot 8 \frac{1}{9} + 72 {(\frac{1}{3})}^{3} + 72 {(\frac{1}{3})}^{4}$

Step 2.1.6.3

Rewrite the expression.

$24 + 8 + 72 {(\frac{1}{3})}^{3} + 72 {(\frac{1}{3})}^{4}$

Step 2.1.7

Apply the product rule to $\frac{1}{3}$ .

$24 + 8 + 72 \frac{1^{3}}{3^{3}} + 72 {(\frac{1}{3})}^{4}$

Step 2.1.8

One to any power is one.

$24 + 8 + 72 \frac{1}{3^{3}} + 72 {(\frac{1}{3})}^{4}$

Step 2.1.9

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

$24 + 8 + 72 (\frac{1}{27}) + 72 {(\frac{1}{3})}^{4}$

Step 2.1.10

Cancel the common factor of $9$ .

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Step 2.1.10.1

Factor $9$ out of $72$ .

$24 + 8 + 9 (8) \frac{1}{27} + 72 {(\frac{1}{3})}^{4}$

Step 2.1.10.2

Factor $9$ out of $27$ .

$24 + 8 + 9 \cdot 8 \frac{1}{9 \cdot 3} + 72 {(\frac{1}{3})}^{4}$

Step 2.1.10.3

Cancel the common factor.

$24 + 8 + 9 \cdot 8 \frac{1}{9 \cdot 3} + 72 {(\frac{1}{3})}^{4}$

Step 2.1.10.4

Rewrite the expression.

$24 + 8 + 8 (\frac{1}{3}) + 72 {(\frac{1}{3})}^{4}$

Step 2.1.11

Combine $8$ and $\frac{1}{3}$ .

$24 + 8 + \frac{8}{3} + 72 {(\frac{1}{3})}^{4}$

Step 2.1.12

Apply the product rule to $\frac{1}{3}$ .

$24 + 8 + \frac{8}{3} + 72 \frac{1^{4}}{3^{4}}$

Step 2.1.13

One to any power is one.

$24 + 8 + \frac{8}{3} + 72 \frac{1}{3^{4}}$

Step 2.1.14

Raise $3$ to the power of $4$ .

$24 + 8 + \frac{8}{3} + 72 (\frac{1}{81})$

Step 2.1.15

Cancel the common factor of $9$ .

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Step 2.1.15.1

Factor $9$ out of $72$ .

$24 + 8 + \frac{8}{3} + 9 (8) \frac{1}{81}$

Step 2.1.15.2

Factor $9$ out of $81$ .

$24 + 8 + \frac{8}{3} + 9 \cdot 8 \frac{1}{9 \cdot 9}$

Step 2.1.15.3

Cancel the common factor.

$24 + 8 + \frac{8}{3} + 9 \cdot 8 \frac{1}{9 \cdot 9}$

Step 2.1.15.4

Rewrite the expression.

$24 + 8 + \frac{8}{3} + 8 (\frac{1}{9})$

Step 2.1.16

Combine $8$ and $\frac{1}{9}$ .

$24 + 8 + \frac{8}{3} + \frac{8}{9}$

Step 2.2

Find the common denominator.

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Step 2.2.1

Write $24$ as a fraction with denominator $1$ .

$\frac{24}{1} + 8 + \frac{8}{3} + \frac{8}{9}$

Step 2.2.2

Multiply $\frac{24}{1}$ by $\frac{9}{9}$ .

$\frac{24}{1} \cdot \frac{9}{9} + 8 + \frac{8}{3} + \frac{8}{9}$

Step 2.2.3

Multiply $\frac{24}{1}$ by $\frac{9}{9}$ .

$\frac{24 \cdot 9}{9} + 8 + \frac{8}{3} + \frac{8}{9}$

Step 2.2.4

Write $8$ as a fraction with denominator $1$ .

$\frac{24 \cdot 9}{9} + \frac{8}{1} + \frac{8}{3} + \frac{8}{9}$

Step 2.2.5

Multiply $\frac{8}{1}$ by $\frac{9}{9}$ .

$\frac{24 \cdot 9}{9} + \frac{8}{1} \cdot \frac{9}{9} + \frac{8}{3} + \frac{8}{9}$

Step 2.2.6

Multiply $\frac{8}{1}$ by $\frac{9}{9}$ .

$\frac{24 \cdot 9}{9} + \frac{8 \cdot 9}{9} + \frac{8}{3} + \frac{8}{9}$

Step 2.2.7

Multiply $\frac{8}{3}$ by $\frac{3}{3}$ .

$\frac{24 \cdot 9}{9} + \frac{8 \cdot 9}{9} + \frac{8}{3} \cdot \frac{3}{3} + \frac{8}{9}$

Step 2.2.8

Multiply $\frac{8}{3}$ by $\frac{3}{3}$ .

$\frac{24 \cdot 9}{9} + \frac{8 \cdot 9}{9} + \frac{8 \cdot 3}{3 \cdot 3} + \frac{8}{9}$

Step 2.2.9

Multiply $3$ by $3$ .

$\frac{24 \cdot 9}{9} + \frac{8 \cdot 9}{9} + \frac{8 \cdot 3}{9} + \frac{8}{9}$

Step 2.3

Combine the numerators over the common denominator.

$\frac{24 \cdot 9 + 8 \cdot 9 + 8 \cdot 3 + 8}{9}$

Step 2.4

Simplify each term.

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Step 2.4.1

Multiply $24$ by $9$ .

$\frac{216 + 8 \cdot 9 + 8 \cdot 3 + 8}{9}$

Step 2.4.2

Multiply $8$ by $9$ .

$\frac{216 + 72 + 8 \cdot 3 + 8}{9}$

Step 2.4.3

Multiply $8$ by $3$ .

$\frac{216 + 72 + 24 + 8}{9}$

Step 2.5

Add $216$ and $72$ .

$\frac{288 + 24 + 8}{9}$

Step 2.6

Add $288$ and $24$ .

$\frac{312 + 8}{9}$

Step 2.7

Add $312$ and $8$ .

$\frac{320}{9}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{320}{9}$

Decimal Form:

$35.\overline{5}$

Mixed Number Form:

$35 \frac{5}{9}$