Algebra Examples

${(4 - 3 i)}^{2} + {(6 + i)}^{2}$

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(4 - 3 i)}^{2}$ as $(4 - 3 i) (4 - 3 i)$ .

$(4 - 3 i) (4 - 3 i) + {(6 + i)}^{2}$

Step 1.2

Expand $(4 - 3 i) (4 - 3 i)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$4 (4 - 3 i) - 3 i (4 - 3 i) + {(6 + i)}^{2}$

Step 1.2.2

Apply the distributive property.

$4 \cdot 4 + 4 (- 3 i) - 3 i (4 - 3 i) + {(6 + i)}^{2}$

Step 1.2.3

Apply the distributive property.

$4 \cdot 4 + 4 (- 3 i) - 3 i \cdot 4 - 3 i (- 3 i) + {(6 + i)}^{2}$

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

Multiply $4$ by $4$ .

$16 + 4 (- 3 i) - 3 i \cdot 4 - 3 i (- 3 i) + {(6 + i)}^{2}$

Step 1.3.1.2

Multiply $- 3$ by $4$ .

$16 - 12 i - 3 i \cdot 4 - 3 i (- 3 i) + {(6 + i)}^{2}$

Step 1.3.1.3

Multiply $4$ by $- 3$ .

$16 - 12 i - 12 i - 3 i (- 3 i) + {(6 + i)}^{2}$

Step 1.3.1.4

Multiply $- 3 i (- 3 i)$ .

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Step 1.3.1.4.1

Multiply $- 3$ by $- 3$ .

$16 - 12 i - 12 i + 9 i i + {(6 + i)}^{2}$

Step 1.3.1.4.2

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

$16 - 12 i - 12 i + 9 (i^{1} i) + {(6 + i)}^{2}$

Step 1.3.1.4.3

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

$16 - 12 i - 12 i + 9 (i^{1} i^{1}) + {(6 + i)}^{2}$

Step 1.3.1.4.4

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

$16 - 12 i - 12 i + 9 i^{1 + 1} + {(6 + i)}^{2}$

Step 1.3.1.4.5

Add $1$ and $1$ .

$16 - 12 i - 12 i + 9 i^{2} + {(6 + i)}^{2}$

Step 1.3.1.5

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

$16 - 12 i - 12 i + 9 \cdot - 1 + {(6 + i)}^{2}$

Step 1.3.1.6

Multiply $9$ by $- 1$ .

$16 - 12 i - 12 i - 9 + {(6 + i)}^{2}$

Step 1.3.2

Subtract $9$ from $16$ .

$7 - 12 i - 12 i + {(6 + i)}^{2}$

Step 1.3.3

Subtract $12 i$ from $- 12 i$ .

$7 - 24 i + {(6 + i)}^{2}$

Step 1.4

Rewrite ${(6 + i)}^{2}$ as $(6 + i) (6 + i)$ .

$7 - 24 i + (6 + i) (6 + i)$

Step 1.5

Expand $(6 + i) (6 + i)$ using the FOIL Method.

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Step 1.5.1

Apply the distributive property.

$7 - 24 i + 6 (6 + i) + i (6 + i)$

Step 1.5.2

Apply the distributive property.

$7 - 24 i + 6 \cdot 6 + 6 i + i (6 + i)$

Step 1.5.3

Apply the distributive property.

$7 - 24 i + 6 \cdot 6 + 6 i + i \cdot 6 + i i$

Step 1.6

Simplify and combine like terms.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

Multiply $6$ by $6$ .

$7 - 24 i + 36 + 6 i + i \cdot 6 + i i$

Step 1.6.1.2

Move $6$ to the left of $i$ .

$7 - 24 i + 36 + 6 i + 6 \cdot i + i i$

Step 1.6.1.3

Multiply $i i$ .

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Step 1.6.1.3.1

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

$7 - 24 i + 36 + 6 i + 6 i + i^{1} i$

Step 1.6.1.3.2

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

$7 - 24 i + 36 + 6 i + 6 i + i^{1} i^{1}$

Step 1.6.1.3.3

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

$7 - 24 i + 36 + 6 i + 6 i + i^{1 + 1}$

Step 1.6.1.3.4

Add $1$ and $1$ .

$7 - 24 i + 36 + 6 i + 6 i + i^{2}$

Step 1.6.1.4

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

$7 - 24 i + 36 + 6 i + 6 i - 1$

Step 1.6.2

Subtract $1$ from $36$ .

$7 - 24 i + 35 + 6 i + 6 i$

Step 1.6.3

Add $6 i$ and $6 i$ .

$7 - 24 i + 35 + 12 i$

Step 2

Simplify by adding terms.

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Step 2.1

Add $7$ and $35$ .

$42 - 24 i + 12 i$

Step 2.2

Add $- 24 i$ and $12 i$ .

$42 - 12 i$