Evaluate (1+5i)/(5-i)

$\frac{1 + 5 i}{5 - i}$

Step 1

Multiply the numerator and denominator of $\frac{1 + 5 i}{5 - i}$ by the conjugate of $5 - i$ to make the denominator real.

$\frac{1 + 5 i}{5 - i} \cdot \frac{5 + i}{5 + i}$

Step 2

Multiply.

Combine.

$\frac{(1 + 5 i) (5 + i)}{(5 - i) (5 + i)}$

Simplify the numerator.

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

Apply the distributive property.

$\frac{1 (5 + i) + 5 i (5 + i)}{(5 - i) (5 + i)}$

Apply the distributive property.

$\frac{1 \cdot 5 + 1 i + 5 i (5 + i)}{(5 - i) (5 + i)}$

Apply the distributive property.

$\frac{1 \cdot 5 + 1 i + 5 i \cdot 5 + 5 i i}{(5 - i) (5 + i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $1$ .

$\frac{5 + 1 i + 5 i \cdot 5 + 5 i i}{(5 - i) (5 + i)}$

Multiply $i$ by $1$ .

$\frac{5 + i + 5 i \cdot 5 + 5 i i}{(5 - i) (5 + i)}$

Multiply $5$ by $5$ .

$\frac{5 + i + 25 i + 5 i i}{(5 - i) (5 + i)}$

Multiply $5 i i$ .

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

$\frac{5 + i + 25 i + 5 (i^{1} i)}{(5 - i) (5 + i)}$

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

$\frac{5 + i + 25 i + 5 (i^{1} i^{1})}{(5 - i) (5 + i)}$

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

$\frac{5 + i + 25 i + 5 i^{1 + 1}}{(5 - i) (5 + i)}$

Add $1$ and $1$ .

$\frac{5 + i + 25 i + 5 i^{2}}{(5 - i) (5 + i)}$

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

$\frac{5 + i + 25 i + 5 \cdot - 1}{(5 - i) (5 + i)}$

Multiply $5$ by $- 1$ .

$\frac{5 + i + 25 i - 5}{(5 - i) (5 + i)}$

Subtract $5$ from $5$ .

$\frac{0 + i + 25 i}{(5 - i) (5 + i)}$

Add $0$ and $i$ .

$\frac{i + 25 i}{(5 - i) (5 + i)}$

Add $i$ and $25 i$ .

$\frac{26 i}{(5 - i) (5 + i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{26 i}{5 (5 + i) - i (5 + i)}$

Apply the distributive property.

$\frac{26 i}{5 \cdot 5 + 5 i - i (5 + i)}$

Apply the distributive property.

$\frac{26 i}{5 \cdot 5 + 5 i - i \cdot 5 - i i}$

Simplify.

Multiply $5$ by $5$ .

$\frac{26 i}{25 + 5 i - i \cdot 5 - i i}$

Multiply $5$ by $- 1$ .

$\frac{26 i}{25 + 5 i - 5 i - i i}$

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

$\frac{26 i}{25 + 5 i - 5 i - (i^{1} i)}$

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

$\frac{26 i}{25 + 5 i - 5 i - (i^{1} i^{1})}$

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

$\frac{26 i}{25 + 5 i - 5 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{26 i}{25 + 5 i - 5 i - i^{2}}$

Subtract $5 i$ from $5 i$ .

$\frac{26 i}{25 + 0 - i^{2}}$

Add $25$ and $0$ .

$\frac{26 i}{25 - i^{2}}$

Simplify each term.

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

$\frac{26 i}{25 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{26 i}{25 + 1}$

Add $25$ and $1$ .

$\frac{26 i}{26}$

Step 3

Cancel the common factor of $26$ .

Cancel the common factor.

$\frac{26 i}{26}$

Divide $i$ by $1$ .

$i$

Evaluate (1+5i)/(5-i)

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