Simplify ((5+i)^2)/(3-i)

$\frac{{(5 + i)}^{2}}{3 - i}$

Multiply the numerator and denominator of $\frac{{(5 + i)}^{2}}{3 - i}$ by the conjugate of $3 - i$ to make the denominator real.

$\frac{{(5 + i)}^{2}}{3 - i} \cdot \frac{3 + i}{3 + i}$

Multiply.

Combine.

$\frac{{(5 + i)}^{2} (3 + i)}{(3 - i) (3 + i)}$

Simplify the numerator.

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

$\frac{(5 + i) (5 + i) (3 + i)}{(3 - i) (3 + i)}$

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

Apply the distributive property.

$\frac{(5 (5 + i) + i (5 + i)) (3 + i)}{(3 - i) (3 + i)}$

Apply the distributive property.

$\frac{(5 \cdot 5 + 5 i + i (5 + i)) (3 + i)}{(3 - i) (3 + i)}$

Apply the distributive property.

$\frac{(5 \cdot 5 + 5 i + i \cdot 5 + i i) (3 + i)}{(3 - i) (3 + i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $5$ .

$\frac{(25 + 5 i + i \cdot 5 + i i) (3 + i)}{(3 - i) (3 + i)}$

Move $5$ to the left of $i$ .

$\frac{(25 + 5 i + 5 \cdot i + i i) (3 + i)}{(3 - i) (3 + i)}$

Multiply $i i$ .

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

$\frac{(25 + 5 i + 5 i + i^{1} i) (3 + i)}{(3 - i) (3 + i)}$

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

$\frac{(25 + 5 i + 5 i + i^{1} i^{1}) (3 + i)}{(3 - i) (3 + i)}$

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

$\frac{(25 + 5 i + 5 i + i^{1 + 1}) (3 + i)}{(3 - i) (3 + i)}$

Add $1$ and $1$ .

$\frac{(25 + 5 i + 5 i + i^{2}) (3 + i)}{(3 - i) (3 + i)}$

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

$\frac{(25 + 5 i + 5 i - 1) (3 + i)}{(3 - i) (3 + i)}$

Subtract $1$ from $25$ .

$\frac{(24 + 5 i + 5 i) (3 + i)}{(3 - i) (3 + i)}$

Add $5 i$ and $5 i$ .

$\frac{(24 + 10 i) (3 + i)}{(3 - i) (3 + i)}$

Expand $(24 + 10 i) (3 + i)$ using the FOIL Method.

Apply the distributive property.

$\frac{24 (3 + i) + 10 i (3 + i)}{(3 - i) (3 + i)}$

Apply the distributive property.

$\frac{24 \cdot 3 + 24 i + 10 i (3 + i)}{(3 - i) (3 + i)}$

Apply the distributive property.

$\frac{24 \cdot 3 + 24 i + 10 i \cdot 3 + 10 i i}{(3 - i) (3 + i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $24$ by $3$ .

$\frac{72 + 24 i + 10 i \cdot 3 + 10 i i}{(3 - i) (3 + i)}$

Multiply $3$ by $10$ .

$\frac{72 + 24 i + 30 i + 10 i i}{(3 - i) (3 + i)}$

Multiply $10 i i$ .

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

$\frac{72 + 24 i + 30 i + 10 (i^{1} i)}{(3 - i) (3 + i)}$

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

$\frac{72 + 24 i + 30 i + 10 (i^{1} i^{1})}{(3 - i) (3 + i)}$

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

$\frac{72 + 24 i + 30 i + 10 i^{1 + 1}}{(3 - i) (3 + i)}$

Add $1$ and $1$ .

$\frac{72 + 24 i + 30 i + 10 i^{2}}{(3 - i) (3 + i)}$

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

$\frac{72 + 24 i + 30 i + 10 \cdot - 1}{(3 - i) (3 + i)}$

Multiply $10$ by $- 1$ .

$\frac{72 + 24 i + 30 i - 10}{(3 - i) (3 + i)}$

Subtract $10$ from $72$ .

$\frac{62 + 24 i + 30 i}{(3 - i) (3 + i)}$

Add $24 i$ and $30 i$ .

$\frac{62 + 54 i}{(3 - i) (3 + i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{62 + 54 i}{3 (3 + i) - i (3 + i)}$

Apply the distributive property.

$\frac{62 + 54 i}{3 \cdot 3 + 3 i - i (3 + i)}$

Apply the distributive property.

$\frac{62 + 54 i}{3 \cdot 3 + 3 i - i \cdot 3 - i i}$

Simplify.

Multiply $3$ by $3$ .

$\frac{62 + 54 i}{9 + 3 i - i \cdot 3 - i i}$

Multiply $3$ by $- 1$ .

$\frac{62 + 54 i}{9 + 3 i - 3 i - i i}$

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

$\frac{62 + 54 i}{9 + 3 i - 3 i - (i^{1} i)}$

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

$\frac{62 + 54 i}{9 + 3 i - 3 i - (i^{1} i^{1})}$

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

$\frac{62 + 54 i}{9 + 3 i - 3 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{62 + 54 i}{9 + 3 i - 3 i - i^{2}}$

Subtract $3 i$ from $3 i$ .

$\frac{62 + 54 i}{9 + 0 - i^{2}}$

Add $9$ and $0$ .

$\frac{62 + 54 i}{9 - i^{2}}$

Simplify each term.

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

$\frac{62 + 54 i}{9 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{62 + 54 i}{9 + 1}$

Add $9$ and $1$ .

$\frac{62 + 54 i}{10}$

Cancel the common factor of $62 + 54 i$ and $10$ .

Factor $2$ out of $62$ .

$\frac{2 (31) + 54 i}{10}$

Factor $2$ out of $54 i$ .

$\frac{2 (31) + 2 (27 i)}{10}$

Factor $2$ out of $2 (31) + 2 (27 i)$ .

$\frac{2 (31 + 27 i)}{10}$

Cancel the common factors.

Factor $2$ out of $10$ .

$\frac{2 (31 + 27 i)}{2 \cdot 5}$

Cancel the common factor.

$\frac{2 (31 + 27 i)}{2 \cdot 5}$

Rewrite the expression.

$\frac{31 + 27 i}{5}$

Split the fraction $\frac{31 + 27 i}{5}$ into two fractions.

$\frac{31}{5} + \frac{27 i}{5}$

Simplify ((5+i)^2)/(3-i)

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