Calculus Examples

$\sum_{i = 2}^{9} {(- \frac{2}{3})}^{2}$

Step 1

Simplify the summation.

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

Apply the product rule to $- \frac{2}{3}$ .

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

Step 1.1.2

Apply the product rule to $\frac{2}{3}$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

$\frac{4}{3^{2}}$

Step 1.5

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

$\frac{4}{9}$

Step 1.6

Rewrite the summation.

$\sum_{i = 2}^{9} \frac{4}{9}$

Step 2

Split the summation to make the starting value of $i$ equal to $1$ .

$\sum_{i = 2}^{9} \frac{4}{9} = \sum_{i = 1}^{9} \frac{4}{9} - \sum_{i = 1}^{1} \frac{4}{9}$

Step 3

Evaluate $\sum_{i = 1}^{9} \frac{4}{9}$ .

Tap for more steps...

Step 3.1

The formula for the summation of a constant is:

$\sum_{k = 1}^{n} c = c n$

Step 3.2

Substitute the values into the formula.

$(\frac{4}{9}) (9)$

Step 3.3

Cancel the common factor of $9$ .

Tap for more steps...

Step 3.3.1

Cancel the common factor.

$\frac{4}{9} \cdot 9$

Step 3.3.2

Rewrite the expression.

$4$

Step 4

Expand the series for each value of $i$ .

$\frac{4}{9}$

Step 5

Replace the summations with the values found.

$4 - \frac{4}{9}$

Step 6

Simplify.

Tap for more steps...

Step 6.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$4 \cdot \frac{9}{9} - \frac{4}{9}$

Step 6.2

Combine $4$ and $\frac{9}{9}$ .

$\frac{4 \cdot 9}{9} - \frac{4}{9}$

Step 6.3

Combine the numerators over the common denominator.

$\frac{4 \cdot 9 - 4}{9}$

Step 6.4

Simplify the numerator.

Tap for more steps...

Step 6.4.1

Multiply $4$ by $9$ .

$\frac{36 - 4}{9}$

Step 6.4.2

Subtract $4$ from $36$ .

$\frac{32}{9}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{32}{9}$

Decimal Form:

$3.\overline{5}$

Mixed Number Form:

$3 \frac{5}{9}$