Calculus Examples

$\sum_{k = 1}^{4} {(k + 8)}^{2}$

Step 1

Simplify the summation.

Tap for more steps...

Step 1.1

Rewrite ${(k + 8)}^{2}$ as $(k + 8) (k + 8)$ .

$(k + 8) (k + 8)$

Step 1.2

Expand $(k + 8) (k + 8)$ using the FOIL Method.

Tap for more steps...

Step 1.2.1

Apply the distributive property.

$k (k + 8) + 8 (k + 8)$

Step 1.2.2

Apply the distributive property.

$k \cdot k + k \cdot 8 + 8 (k + 8)$

Step 1.2.3

Apply the distributive property.

$k \cdot k + k \cdot 8 + 8 k + 8 \cdot 8$

Step 1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.3.1

Simplify each term.

Tap for more steps...

Step 1.3.1.1

Multiply $k$ by $k$ .

$k^{2} + k \cdot 8 + 8 k + 8 \cdot 8$

Step 1.3.1.2

Move $8$ to the left of $k$ .

$k^{2} + 8 \cdot k + 8 k + 8 \cdot 8$

Step 1.3.1.3

Multiply $8$ by $8$ .

$k^{2} + 8 k + 8 k + 64$

Step 1.3.2

Add $8 k$ and $8 k$ .

$k^{2} + 16 k + 64$

Step 1.4

Rewrite the summation.

$\sum_{k = 1}^{4} k^{2} + 16 k + 64$

Step 2

Expand the series for each value of $k$ .

$1^{2} + 16 \cdot 1 + 64 + 2^{2} + 16 \cdot 2 + 64 + 3^{2} + 16 \cdot 3 + 64 + 4^{2} + 16 \cdot 4 + 64$

Step 3

Simplify.

Tap for more steps...

Step 3.1

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

$1 + 16 \cdot 1 + 64 + 2^{2} + 16 \cdot 2 + 64 + 3^{2} + 16 \cdot 3 + 64 + 4^{2} + 16 \cdot 4 + 64$

Step 3.2

Multiply $16$ by $1$ .

$1 + 16 + 64 + 2^{2} + 16 \cdot 2 + 64 + 3^{2} + 16 \cdot 3 + 64 + 4^{2} + 16 \cdot 4 + 64$

Step 3.3

Add $1$ and $16$ .

$17 + 64 + 2^{2} + 16 \cdot 2 + 64 + 3^{2} + 16 \cdot 3 + 64 + 4^{2} + 16 \cdot 4 + 64$

Step 3.4

Add $17$ and $64$ .

$81 + 2^{2} + 16 \cdot 2 + 64 + 3^{2} + 16 \cdot 3 + 64 + 4^{2} + 16 \cdot 4 + 64$

Step 3.5

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

$81 + 4 + 16 \cdot 2 + 64 + 3^{2} + 16 \cdot 3 + 64 + 4^{2} + 16 \cdot 4 + 64$

Step 3.6

Multiply $16$ by $2$ .

$81 + 4 + 32 + 64 + 3^{2} + 16 \cdot 3 + 64 + 4^{2} + 16 \cdot 4 + 64$

Step 3.7

Add $4$ and $32$ .

$81 + 36 + 64 + 3^{2} + 16 \cdot 3 + 64 + 4^{2} + 16 \cdot 4 + 64$

Step 3.8

Add $36$ and $64$ .

$81 + 100 + 3^{2} + 16 \cdot 3 + 64 + 4^{2} + 16 \cdot 4 + 64$

Step 3.9

Add $81$ and $100$ .

$181 + 3^{2} + 16 \cdot 3 + 64 + 4^{2} + 16 \cdot 4 + 64$

Step 3.10

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

$181 + 9 + 16 \cdot 3 + 64 + 4^{2} + 16 \cdot 4 + 64$

Step 3.11

Multiply $16$ by $3$ .

$181 + 9 + 48 + 64 + 4^{2} + 16 \cdot 4 + 64$

Step 3.12

Add $9$ and $48$ .

$181 + 57 + 64 + 4^{2} + 16 \cdot 4 + 64$

Step 3.13

Add $57$ and $64$ .

$181 + 121 + 4^{2} + 16 \cdot 4 + 64$

Step 3.14

Add $181$ and $121$ .

$302 + 4^{2} + 16 \cdot 4 + 64$

Step 3.15

Raise $4$ to the power of $2$ .

$302 + 16 + 16 \cdot 4 + 64$

Step 3.16

Multiply $16$ by $4$ .

$302 + 16 + 64 + 64$

Step 3.17

Add $16$ and $64$ .

$302 + 80 + 64$

Step 3.18

Add $80$ and $64$ .

$302 + 144$

Step 3.19

Add $302$ and $144$ .

$446$