Calculus Examples

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

Step 1

Simplify the summation.

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Step 1.1

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

$(2 i j + 8 j) (2 i j + 8 j)$

Step 1.2

Expand $(2 i j + 8 j) (2 i j + 8 j)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$2 i j (2 i j + 8 j) + 8 j (2 i j + 8 j)$

Step 1.2.2

Apply the distributive property.

$2 i j (2 i j) + 2 i j (8 j) + 8 j (2 i j + 8 j)$

Step 1.2.3

Apply the distributive property.

$2 i j (2 i j) + 2 i j (8 j) + 8 j (2 i j) + 8 j (8 j)$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Rewrite using the commutative property of multiplication.

$2 i \cdot 2 i j \cdot j + 2 i j (8 j) + 8 j (2 i j) + 8 j (8 j)$

Step 1.3.1.2

Multiply $j$ by $j$ by adding the exponents.

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Step 1.3.1.2.1

Move $j$ .

$2 i \cdot 2 i (j \cdot j) + 2 i j (8 j) + 8 j (2 i j) + 8 j (8 j)$

Step 1.3.1.2.2

Multiply $j$ by $j$ .

$2 i \cdot 2 i j^{2} + 2 i j (8 j) + 8 j (2 i j) + 8 j (8 j)$

Step 1.3.1.3

Multiply $2$ by $2$ .

$4 i i j^{2} + 2 i j (8 j) + 8 j (2 i j) + 8 j (8 j)$

Step 1.3.1.4

Multiply $4 i i$ .

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Step 1.3.1.4.1

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

$4 (i^{1} i) j^{2} + 2 i j (8 j) + 8 j (2 i j) + 8 j (8 j)$

Step 1.3.1.4.2

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

$4 (i^{1} i^{1}) j^{2} + 2 i j (8 j) + 8 j (2 i j) + 8 j (8 j)$

Step 1.3.1.4.3

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

$4 i^{1 + 1} j^{2} + 2 i j (8 j) + 8 j (2 i j) + 8 j (8 j)$

Step 1.3.1.4.4

Add $1$ and $1$ .

$4 i^{2} j^{2} + 2 i j (8 j) + 8 j (2 i j) + 8 j (8 j)$

Step 1.3.1.5

Rewrite using the commutative property of multiplication.

$4 i^{2} j^{2} + 2 i \cdot 8 j \cdot j + 8 j (2 i j) + 8 j (8 j)$

Step 1.3.1.6

Multiply $j$ by $j$ by adding the exponents.

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Step 1.3.1.6.1

Move $j$ .

$4 i^{2} j^{2} + 2 i \cdot 8 (j \cdot j) + 8 j (2 i j) + 8 j (8 j)$

Step 1.3.1.6.2

Multiply $j$ by $j$ .

$4 i^{2} j^{2} + 2 i \cdot 8 j^{2} + 8 j (2 i j) + 8 j (8 j)$

Step 1.3.1.7

Multiply $8$ by $2$ .

$4 i^{2} j^{2} + 16 i j^{2} + 8 j (2 i j) + 8 j (8 j)$

Step 1.3.1.8

Rewrite using the commutative property of multiplication.

$4 i^{2} j^{2} + 16 i j^{2} + 8 \cdot 2 i j \cdot j + 8 j (8 j)$

Step 1.3.1.9

Multiply $j$ by $j$ by adding the exponents.

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Step 1.3.1.9.1

Move $j$ .

$4 i^{2} j^{2} + 16 i j^{2} + 8 \cdot 2 i (j \cdot j) + 8 j (8 j)$

Step 1.3.1.9.2

Multiply $j$ by $j$ .

$4 i^{2} j^{2} + 16 i j^{2} + 8 \cdot 2 i j^{2} + 8 j (8 j)$

Step 1.3.1.10

Multiply $8$ by $2$ .

$4 i^{2} j^{2} + 16 i j^{2} + 16 i j^{2} + 8 j (8 j)$

Step 1.3.1.11

Rewrite using the commutative property of multiplication.

$4 i^{2} j^{2} + 16 i j^{2} + 16 i j^{2} + 8 \cdot 8 j \cdot j$

Step 1.3.1.12

Multiply $j$ by $j$ by adding the exponents.

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Step 1.3.1.12.1

Move $j$ .

$4 i^{2} j^{2} + 16 i j^{2} + 16 i j^{2} + 8 \cdot 8 (j \cdot j)$

Step 1.3.1.12.2

Multiply $j$ by $j$ .

$4 i^{2} j^{2} + 16 i j^{2} + 16 i j^{2} + 8 \cdot 8 j^{2}$

Step 1.3.1.13

Multiply $8$ by $8$ .

$4 i^{2} j^{2} + 16 i j^{2} + 16 i j^{2} + 64 j^{2}$

Step 1.3.2

Add $16 i j^{2}$ and $16 i j^{2}$ .

$4 i^{2} j^{2} + 32 i j^{2} + 64 j^{2}$

Step 1.4

Reorder factors in $4 i^{2} j^{2} + 32 i j^{2} + 64 j^{2}$ .

$4 j^{2} i^{2} + 32 j^{2} i + 64 j^{2}$

Step 1.5

Rewrite the summation.

$\sum_{j = 1}^{4} 4 j^{2} i^{2} + 32 j^{2} i + 64 j^{2}$

Step 2

Expand the series for each value of $j$ .

$4 \cdot 1^{2} i^{2} + 32 \cdot 1^{2} i + 64 \cdot 1^{2} + 4 \cdot 2^{2} i^{2} + 32 \cdot 2^{2} i + 64 \cdot 2^{2} + 4 \cdot 3^{2} i^{2} + 32 \cdot 3^{2} i + 64 \cdot 3^{2} + 4 \cdot 4^{2} i^{2} + 32 \cdot 4^{2} i + 64 \cdot 4^{2}$

Step 3

Substitute the values back and simplify $4 \cdot (1^{2} i^{2}) + 32 \cdot (1^{2} i) + 64 \cdot 1^{2} + 4 \cdot (2^{2} i^{2}) + 32 \cdot (2^{2} i) + 64 \cdot 2^{2} + 4 \cdot (3^{2} i^{2}) + 32 \cdot (3^{2} i) + 64 \cdot 3^{2} + 4 \cdot (4^{2} i^{2}) + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$ .

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Step 3.1

Simplify each term.

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Step 3.1.1

One to any power is one.

$4 \cdot (1 i^{2}) + 32 \cdot (1^{2} i) + 64 \cdot 1^{2} + 4 \cdot (2^{2} i^{2}) + 32 \cdot (2^{2} i) + 64 \cdot 2^{2} + 4 \cdot (3^{2} i^{2}) + 32 \cdot (3^{2} i) + 64 \cdot 3^{2} + 4 \cdot (4^{2} i^{2}) + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.2

Multiply $i^{2}$ by $1$ .

$4 \cdot i^{2} + 32 \cdot (1^{2} i) + 64 \cdot 1^{2} + 4 \cdot (2^{2} i^{2}) + 32 \cdot (2^{2} i) + 64 \cdot 2^{2} + 4 \cdot (3^{2} i^{2}) + 32 \cdot (3^{2} i) + 64 \cdot 3^{2} + 4 \cdot (4^{2} i^{2}) + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.3

Rewrite $i^{2}$ as $- 1$ .

$4 \cdot - 1 + 32 \cdot (1^{2} i) + 64 \cdot 1^{2} + 4 \cdot (2^{2} i^{2}) + 32 \cdot (2^{2} i) + 64 \cdot 2^{2} + 4 \cdot (3^{2} i^{2}) + 32 \cdot (3^{2} i) + 64 \cdot 3^{2} + 4 \cdot (4^{2} i^{2}) + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.4

Multiply $4$ by $- 1$ .

$- 4 + 32 \cdot (1^{2} i) + 64 \cdot 1^{2} + 4 \cdot (2^{2} i^{2}) + 32 \cdot (2^{2} i) + 64 \cdot 2^{2} + 4 \cdot (3^{2} i^{2}) + 32 \cdot (3^{2} i) + 64 \cdot 3^{2} + 4 \cdot (4^{2} i^{2}) + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.5

One to any power is one.

$- 4 + 32 \cdot (1 i) + 64 \cdot 1^{2} + 4 \cdot (2^{2} i^{2}) + 32 \cdot (2^{2} i) + 64 \cdot 2^{2} + 4 \cdot (3^{2} i^{2}) + 32 \cdot (3^{2} i) + 64 \cdot 3^{2} + 4 \cdot (4^{2} i^{2}) + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.6

Multiply $i$ by $1$ .

$- 4 + 32 \cdot i + 64 \cdot 1^{2} + 4 \cdot (2^{2} i^{2}) + 32 \cdot (2^{2} i) + 64 \cdot 2^{2} + 4 \cdot (3^{2} i^{2}) + 32 \cdot (3^{2} i) + 64 \cdot 3^{2} + 4 \cdot (4^{2} i^{2}) + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.7

One to any power is one.

$- 4 + 32 i + 64 \cdot 1 + 4 \cdot (2^{2} i^{2}) + 32 \cdot (2^{2} i) + 64 \cdot 2^{2} + 4 \cdot (3^{2} i^{2}) + 32 \cdot (3^{2} i) + 64 \cdot 3^{2} + 4 \cdot (4^{2} i^{2}) + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.8

Multiply $64$ by $1$ .

$- 4 + 32 i + 64 + 4 \cdot (2^{2} i^{2}) + 32 \cdot (2^{2} i) + 64 \cdot 2^{2} + 4 \cdot (3^{2} i^{2}) + 32 \cdot (3^{2} i) + 64 \cdot 3^{2} + 4 \cdot (4^{2} i^{2}) + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.9

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

$- 4 + 32 i + 64 + 4 \cdot (4 i^{2}) + 32 \cdot (2^{2} i) + 64 \cdot 2^{2} + 4 \cdot (3^{2} i^{2}) + 32 \cdot (3^{2} i) + 64 \cdot 3^{2} + 4 \cdot (4^{2} i^{2}) + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.10

Rewrite $i^{2}$ as $- 1$ .

$- 4 + 32 i + 64 + 4 \cdot (4 \cdot - 1) + 32 \cdot (2^{2} i) + 64 \cdot 2^{2} + 4 \cdot (3^{2} i^{2}) + 32 \cdot (3^{2} i) + 64 \cdot 3^{2} + 4 \cdot (4^{2} i^{2}) + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.11

Multiply $4 (4 \cdot - 1)$ .

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Step 3.1.11.1

Multiply $4$ by $- 1$ .

$- 4 + 32 i + 64 + 4 \cdot - 4 + 32 \cdot (2^{2} i) + 64 \cdot 2^{2} + 4 \cdot (3^{2} i^{2}) + 32 \cdot (3^{2} i) + 64 \cdot 3^{2} + 4 \cdot (4^{2} i^{2}) + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.11.2

Multiply $4$ by $- 4$ .

$- 4 + 32 i + 64 - 16 + 32 \cdot (2^{2} i) + 64 \cdot 2^{2} + 4 \cdot (3^{2} i^{2}) + 32 \cdot (3^{2} i) + 64 \cdot 3^{2} + 4 \cdot (4^{2} i^{2}) + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.12

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

$- 4 + 32 i + 64 - 16 + 32 \cdot (4 i) + 64 \cdot 2^{2} + 4 \cdot (3^{2} i^{2}) + 32 \cdot (3^{2} i) + 64 \cdot 3^{2} + 4 \cdot (4^{2} i^{2}) + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.13

Multiply $4$ by $32$ .

$- 4 + 32 i + 64 - 16 + 128 i + 64 \cdot 2^{2} + 4 \cdot (3^{2} i^{2}) + 32 \cdot (3^{2} i) + 64 \cdot 3^{2} + 4 \cdot (4^{2} i^{2}) + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.14

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

$- 4 + 32 i + 64 - 16 + 128 i + 64 \cdot 4 + 4 \cdot (3^{2} i^{2}) + 32 \cdot (3^{2} i) + 64 \cdot 3^{2} + 4 \cdot (4^{2} i^{2}) + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.15

Multiply $64$ by $4$ .

$- 4 + 32 i + 64 - 16 + 128 i + 256 + 4 \cdot (3^{2} i^{2}) + 32 \cdot (3^{2} i) + 64 \cdot 3^{2} + 4 \cdot (4^{2} i^{2}) + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.16

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

$- 4 + 32 i + 64 - 16 + 128 i + 256 + 4 \cdot (9 i^{2}) + 32 \cdot (3^{2} i) + 64 \cdot 3^{2} + 4 \cdot (4^{2} i^{2}) + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.17

Rewrite $i^{2}$ as $- 1$ .

$- 4 + 32 i + 64 - 16 + 128 i + 256 + 4 \cdot (9 \cdot - 1) + 32 \cdot (3^{2} i) + 64 \cdot 3^{2} + 4 \cdot (4^{2} i^{2}) + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.18

Multiply $4 (9 \cdot - 1)$ .

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Step 3.1.18.1

Multiply $9$ by $- 1$ .

$- 4 + 32 i + 64 - 16 + 128 i + 256 + 4 \cdot - 9 + 32 \cdot (3^{2} i) + 64 \cdot 3^{2} + 4 \cdot (4^{2} i^{2}) + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.18.2

Multiply $4$ by $- 9$ .

$- 4 + 32 i + 64 - 16 + 128 i + 256 - 36 + 32 \cdot (3^{2} i) + 64 \cdot 3^{2} + 4 \cdot (4^{2} i^{2}) + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.19

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

$- 4 + 32 i + 64 - 16 + 128 i + 256 - 36 + 32 \cdot (9 i) + 64 \cdot 3^{2} + 4 \cdot (4^{2} i^{2}) + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.20

Multiply $9$ by $32$ .

$- 4 + 32 i + 64 - 16 + 128 i + 256 - 36 + 288 i + 64 \cdot 3^{2} + 4 \cdot (4^{2} i^{2}) + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.21

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

$- 4 + 32 i + 64 - 16 + 128 i + 256 - 36 + 288 i + 64 \cdot 9 + 4 \cdot (4^{2} i^{2}) + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.22

Multiply $64$ by $9$ .

$- 4 + 32 i + 64 - 16 + 128 i + 256 - 36 + 288 i + 576 + 4 \cdot (4^{2} i^{2}) + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.23

Multiply $4$ by $4^{2}$ by adding the exponents.

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Step 3.1.23.1

Move $4^{2}$ .

$- 4 + 32 i + 64 - 16 + 128 i + 256 - 36 + 288 i + 576 + 4^{2} \cdot 4 \cdot i^{2} + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.23.2

Multiply $4^{2}$ by $4$ .

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Step 3.1.23.2.1

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

$- 4 + 32 i + 64 - 16 + 128 i + 256 - 36 + 288 i + 576 + 4^{2} \cdot 4^{1} \cdot i^{2} + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.23.2.2

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

$- 4 + 32 i + 64 - 16 + 128 i + 256 - 36 + 288 i + 576 + 4^{2 + 1} \cdot i^{2} + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.23.3

Add $2$ and $1$ .

$- 4 + 32 i + 64 - 16 + 128 i + 256 - 36 + 288 i + 576 + 4^{3} \cdot i^{2} + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.24

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

$- 4 + 32 i + 64 - 16 + 128 i + 256 - 36 + 288 i + 576 + 64 \cdot i^{2} + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.25

Rewrite $i^{2}$ as $- 1$ .

$- 4 + 32 i + 64 - 16 + 128 i + 256 - 36 + 288 i + 576 + 64 \cdot - 1 + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.26

Multiply $64$ by $- 1$ .

$- 4 + 32 i + 64 - 16 + 128 i + 256 - 36 + 288 i + 576 - 64 + 32 \cdot (4^{2} i) + 64 \cdot 4^{2}$

Step 3.1.27

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

$- 4 + 32 i + 64 - 16 + 128 i + 256 - 36 + 288 i + 576 - 64 + 32 \cdot (16 i) + 64 \cdot 4^{2}$

Step 3.1.28

Multiply $16$ by $32$ .

$- 4 + 32 i + 64 - 16 + 128 i + 256 - 36 + 288 i + 576 - 64 + 512 i + 64 \cdot 4^{2}$

Step 3.1.29

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

$- 4 + 32 i + 64 - 16 + 128 i + 256 - 36 + 288 i + 576 - 64 + 512 i + 64 \cdot 16$

Step 3.1.30

Multiply $64$ by $16$ .

$- 4 + 32 i + 64 - 16 + 128 i + 256 - 36 + 288 i + 576 - 64 + 512 i + 1024$

Step 3.2

Simplify by adding terms.

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Step 3.2.1

Add $- 4$ and $64$ .

$60 + 32 i - 16 + 128 i + 256 - 36 + 288 i + 576 - 64 + 512 i + 1024$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Subtract $16$ from $60$ .

$44 + 32 i + 128 i + 256 - 36 + 288 i + 576 - 64 + 512 i + 1024$

Step 3.2.2.2

Add $44$ and $256$ .

$300 + 32 i + 128 i - 36 + 288 i + 576 - 64 + 512 i + 1024$

Step 3.2.2.3

Subtract $36$ from $300$ .

$264 + 32 i + 128 i + 288 i + 576 - 64 + 512 i + 1024$

Step 3.2.2.4

Add $264$ and $576$ .

$840 + 32 i + 128 i + 288 i - 64 + 512 i + 1024$

Step 3.2.2.5

Subtract $64$ from $840$ .

$776 + 32 i + 128 i + 288 i + 512 i + 1024$

Step 3.2.2.6

Add $776$ and $1024$ .

$1800 + 32 i + 128 i + 288 i + 512 i$

Step 3.2.3

Add $32 i$ and $128 i$ .

$1800 + 160 i + 288 i + 512 i$

Step 3.2.4

Add $160 i$ and $288 i$ .

$1800 + 448 i + 512 i$

Step 3.2.5

Add $448 i$ and $512 i$ .

$1800 + 960 i$