Algebra Examples

$\frac{6 + i}{6 - i}$

Step 1

Multiply the numerator and denominator of $\frac{6 + i}{6 - i}$ by the conjugate of $6 - i$ to make the denominator real.

$\frac{6 + i}{6 - i} \cdot \frac{6 + i}{6 + i}$

Step 2

Multiply.

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Step 2.1

Combine.

$\frac{(6 + i) (6 + i)}{(6 - i) (6 + i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

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

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Step 2.2.1.1

Apply the distributive property.

$\frac{6 (6 + i) + i (6 + i)}{(6 - i) (6 + i)}$

Step 2.2.1.2

Apply the distributive property.

$\frac{6 \cdot 6 + 6 i + i (6 + i)}{(6 - i) (6 + i)}$

Step 2.2.1.3

Apply the distributive property.

$\frac{6 \cdot 6 + 6 i + i \cdot 6 + i i}{(6 - i) (6 + i)}$

Step 2.2.2

Simplify and combine like terms.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Multiply $6$ by $6$ .

$\frac{36 + 6 i + i \cdot 6 + i i}{(6 - i) (6 + i)}$

Step 2.2.2.1.2

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

$\frac{36 + 6 i + 6 \cdot i + i i}{(6 - i) (6 + i)}$

Step 2.2.2.1.3

Multiply $i i$ .

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Step 2.2.2.1.3.1

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

$\frac{36 + 6 i + 6 i + i^{1} i}{(6 - i) (6 + i)}$

Step 2.2.2.1.3.2

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

$\frac{36 + 6 i + 6 i + i^{1} i^{1}}{(6 - i) (6 + i)}$

Step 2.2.2.1.3.3

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

$\frac{36 + 6 i + 6 i + i^{1 + 1}}{(6 - i) (6 + i)}$

Step 2.2.2.1.3.4

Add $1$ and $1$ .

$\frac{36 + 6 i + 6 i + i^{2}}{(6 - i) (6 + i)}$

Step 2.2.2.1.4

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

$\frac{36 + 6 i + 6 i - 1}{(6 - i) (6 + i)}$

Step 2.2.2.2

Subtract $1$ from $36$ .

$\frac{35 + 6 i + 6 i}{(6 - i) (6 + i)}$

Step 2.2.2.3

Add $6 i$ and $6 i$ .

$\frac{35 + 12 i}{(6 - i) (6 + i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

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

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Step 2.3.1.1

Apply the distributive property.

$\frac{35 + 12 i}{6 (6 + i) - i (6 + i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{35 + 12 i}{6 \cdot 6 + 6 i - i (6 + i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{35 + 12 i}{6 \cdot 6 + 6 i - i \cdot 6 - i i}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $6$ by $6$ .

$\frac{35 + 12 i}{36 + 6 i - i \cdot 6 - i i}$

Step 2.3.2.2

Multiply $6$ by $- 1$ .

$\frac{35 + 12 i}{36 + 6 i - 6 i - i i}$

Step 2.3.2.3

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

$\frac{35 + 12 i}{36 + 6 i - 6 i - (i^{1} i)}$

Step 2.3.2.4

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

$\frac{35 + 12 i}{36 + 6 i - 6 i - (i^{1} i^{1})}$

Step 2.3.2.5

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

$\frac{35 + 12 i}{36 + 6 i - 6 i - i^{1 + 1}}$

Step 2.3.2.6

Add $1$ and $1$ .

$\frac{35 + 12 i}{36 + 6 i - 6 i - i^{2}}$

Step 2.3.2.7

Subtract $6 i$ from $6 i$ .

$\frac{35 + 12 i}{36 + 0 - i^{2}}$

Step 2.3.2.8

Add $36$ and $0$ .

$\frac{35 + 12 i}{36 - i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{35 + 12 i}{36 - - 1}$

Step 2.3.3.2

Multiply $- 1$ by $- 1$ .

$\frac{35 + 12 i}{36 + 1}$

Step 2.3.4

Add $36$ and $1$ .

$\frac{35 + 12 i}{37}$

Step 3

Split the fraction $\frac{35 + 12 i}{37}$ into two fractions.

$\frac{35}{37} + \frac{12 i}{37}$