Write in Standard Form (4+7i)/(7-i)

$\frac{4 + 7 i}{7 - i}$

Step 1

Multiply the numerator and denominator of $\frac{4 + 7 i}{7 - i}$ by the conjugate of $7 - i$ to make the denominator real.

$\frac{4 + 7 i}{7 - i} \cdot \frac{7 + i}{7 + i}$

Step 2

Multiply.

Combine.

$\frac{(4 + 7 i) (7 + i)}{(7 - i) (7 + i)}$

Simplify the numerator.

Expand $(4 + 7 i) (7 + i)$ using the FOIL Method.

Apply the distributive property.

$\frac{4 (7 + i) + 7 i (7 + i)}{(7 - i) (7 + i)}$

Apply the distributive property.

$\frac{4 \cdot 7 + 4 i + 7 i (7 + i)}{(7 - i) (7 + i)}$

Apply the distributive property.

$\frac{4 \cdot 7 + 4 i + 7 i \cdot 7 + 7 i i}{(7 - i) (7 + i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $7$ .

$\frac{28 + 4 i + 7 i \cdot 7 + 7 i i}{(7 - i) (7 + i)}$

Multiply $7$ by $7$ .

$\frac{28 + 4 i + 49 i + 7 i i}{(7 - i) (7 + i)}$

Multiply $7 i i$ .

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

$\frac{28 + 4 i + 49 i + 7 (i^{1} i)}{(7 - i) (7 + i)}$

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

$\frac{28 + 4 i + 49 i + 7 (i^{1} i^{1})}{(7 - i) (7 + i)}$

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

$\frac{28 + 4 i + 49 i + 7 i^{1 + 1}}{(7 - i) (7 + i)}$

Add $1$ and $1$ .

$\frac{28 + 4 i + 49 i + 7 i^{2}}{(7 - i) (7 + i)}$

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

$\frac{28 + 4 i + 49 i + 7 \cdot - 1}{(7 - i) (7 + i)}$

Multiply $7$ by $- 1$ .

$\frac{28 + 4 i + 49 i - 7}{(7 - i) (7 + i)}$

Subtract $7$ from $28$ .

$\frac{21 + 4 i + 49 i}{(7 - i) (7 + i)}$

Add $4 i$ and $49 i$ .

$\frac{21 + 53 i}{(7 - i) (7 + i)}$

Simplify the denominator.

Expand $(7 - i) (7 + i)$ using the FOIL Method.

Apply the distributive property.

$\frac{21 + 53 i}{7 (7 + i) - i (7 + i)}$

Apply the distributive property.

$\frac{21 + 53 i}{7 \cdot 7 + 7 i - i (7 + i)}$

Apply the distributive property.

$\frac{21 + 53 i}{7 \cdot 7 + 7 i - i \cdot 7 - i i}$

Simplify.

Multiply $7$ by $7$ .

$\frac{21 + 53 i}{49 + 7 i - i \cdot 7 - i i}$

Multiply $7$ by $- 1$ .

$\frac{21 + 53 i}{49 + 7 i - 7 i - i i}$

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

$\frac{21 + 53 i}{49 + 7 i - 7 i - (i^{1} i)}$

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

$\frac{21 + 53 i}{49 + 7 i - 7 i - (i^{1} i^{1})}$

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

$\frac{21 + 53 i}{49 + 7 i - 7 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{21 + 53 i}{49 + 7 i - 7 i - i^{2}}$

Subtract $7 i$ from $7 i$ .

$\frac{21 + 53 i}{49 + 0 - i^{2}}$

Add $49$ and $0$ .

$\frac{21 + 53 i}{49 - i^{2}}$

Simplify each term.

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

$\frac{21 + 53 i}{49 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{21 + 53 i}{49 + 1}$

Add $49$ and $1$ .

$\frac{21 + 53 i}{50}$

Step 3

Split the fraction $\frac{21 + 53 i}{50}$ into two fractions.

$\frac{21}{50} + \frac{53 i}{50}$

Write in Standard Form (4+7i)/(7-i)

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