Basic Math Examples

${(5 + 4 i)}^{2} - {(1 - 5 i)}^{2}$

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(5 + 4 i)}^{2}$ as $(5 + 4 i) (5 + 4 i)$ .

$(5 + 4 i) (5 + 4 i) - {(1 - 5 i)}^{2}$

Step 1.2

Expand $(5 + 4 i) (5 + 4 i)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$5 (5 + 4 i) + 4 i (5 + 4 i) - {(1 - 5 i)}^{2}$

Step 1.2.2

Apply the distributive property.

$5 \cdot 5 + 5 (4 i) + 4 i (5 + 4 i) - {(1 - 5 i)}^{2}$

Step 1.2.3

Apply the distributive property.

$5 \cdot 5 + 5 (4 i) + 4 i \cdot 5 + 4 i (4 i) - {(1 - 5 i)}^{2}$

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 $5$ by $5$ .

$25 + 5 (4 i) + 4 i \cdot 5 + 4 i (4 i) - {(1 - 5 i)}^{2}$

Step 1.3.1.2

Multiply $4$ by $5$ .

$25 + 20 i + 4 i \cdot 5 + 4 i (4 i) - {(1 - 5 i)}^{2}$

Step 1.3.1.3

Multiply $5$ by $4$ .

$25 + 20 i + 20 i + 4 i (4 i) - {(1 - 5 i)}^{2}$

Step 1.3.1.4

Multiply $4 i (4 i)$ .

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Step 1.3.1.4.1

Multiply $4$ by $4$ .

$25 + 20 i + 20 i + 16 i i - {(1 - 5 i)}^{2}$

Step 1.3.1.4.2

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

$25 + 20 i + 20 i + 16 (i^{1} i) - {(1 - 5 i)}^{2}$

Step 1.3.1.4.3

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

$25 + 20 i + 20 i + 16 (i^{1} i^{1}) - {(1 - 5 i)}^{2}$

Step 1.3.1.4.4

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

$25 + 20 i + 20 i + 16 i^{1 + 1} - {(1 - 5 i)}^{2}$

Step 1.3.1.4.5

Add $1$ and $1$ .

$25 + 20 i + 20 i + 16 i^{2} - {(1 - 5 i)}^{2}$

Step 1.3.1.5

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

$25 + 20 i + 20 i + 16 \cdot - 1 - {(1 - 5 i)}^{2}$

Step 1.3.1.6

Multiply $16$ by $- 1$ .

$25 + 20 i + 20 i - 16 - {(1 - 5 i)}^{2}$

Step 1.3.2

Subtract $16$ from $25$ .

$9 + 20 i + 20 i - {(1 - 5 i)}^{2}$

Step 1.3.3

Add $20 i$ and $20 i$ .

$9 + 40 i - {(1 - 5 i)}^{2}$

Step 1.4

Rewrite ${(1 - 5 i)}^{2}$ as $(1 - 5 i) (1 - 5 i)$ .

$9 + 40 i - ((1 - 5 i) (1 - 5 i))$

Step 1.5

Expand $(1 - 5 i) (1 - 5 i)$ using the FOIL Method.

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Step 1.5.1

Apply the distributive property.

$9 + 40 i - (1 (1 - 5 i) - 5 i (1 - 5 i))$

Step 1.5.2

Apply the distributive property.

$9 + 40 i - (1 \cdot 1 + 1 (- 5 i) - 5 i (1 - 5 i))$

Step 1.5.3

Apply the distributive property.

$9 + 40 i - (1 \cdot 1 + 1 (- 5 i) - 5 i \cdot 1 - 5 i (- 5 i))$

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 $1$ by $1$ .

$9 + 40 i - (1 + 1 (- 5 i) - 5 i \cdot 1 - 5 i (- 5 i))$

Step 1.6.1.2

Multiply $- 5 i$ by $1$ .

$9 + 40 i - (1 - 5 i - 5 i \cdot 1 - 5 i (- 5 i))$

Step 1.6.1.3

Multiply $- 5$ by $1$ .

$9 + 40 i - (1 - 5 i - 5 i - 5 i (- 5 i))$

Step 1.6.1.4

Multiply $- 5 i (- 5 i)$ .

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Step 1.6.1.4.1

Multiply $- 5$ by $- 5$ .

$9 + 40 i - (1 - 5 i - 5 i + 25 i i)$

Step 1.6.1.4.2

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

$9 + 40 i - (1 - 5 i - 5 i + 25 (i^{1} i))$

Step 1.6.1.4.3

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

$9 + 40 i - (1 - 5 i - 5 i + 25 (i^{1} i^{1}))$

Step 1.6.1.4.4

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

$9 + 40 i - (1 - 5 i - 5 i + 25 i^{1 + 1})$

Step 1.6.1.4.5

Add $1$ and $1$ .

$9 + 40 i - (1 - 5 i - 5 i + 25 i^{2})$

Step 1.6.1.5

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

$9 + 40 i - (1 - 5 i - 5 i + 25 \cdot - 1)$

Step 1.6.1.6

Multiply $25$ by $- 1$ .

$9 + 40 i - (1 - 5 i - 5 i - 25)$

Step 1.6.2

Subtract $25$ from $1$ .

$9 + 40 i - (- 24 - 5 i - 5 i)$

Step 1.6.3

Subtract $5 i$ from $- 5 i$ .

$9 + 40 i - (- 24 - 10 i)$

Step 1.7

Apply the distributive property.

$9 + 40 i - - 24 - (- 10 i)$

Step 1.8

Multiply $- 1$ by $- 24$ .

$9 + 40 i + 24 - (- 10 i)$

Step 1.9

Multiply $- 10$ by $- 1$ .

$9 + 40 i + 24 + 10 i$

Step 2

Simplify by adding terms.

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Step 2.1

Add $9$ and $24$ .

$33 + 40 i + 10 i$

Step 2.2

Add $40 i$ and $10 i$ .

$33 + 50 i$