Algebra Examples

$\frac{7 + 5 i}{- 7 i}$

Step 1

Multiply the numerator and denominator of $\frac{7 + 5 i}{- 7 i}$ by the conjugate of $- 7 i$ to make the denominator real.

$\frac{7 + 5 i}{- 7 i} \cdot \frac{i}{i}$

Step 2

Multiply.

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Step 2.1

Combine.

$\frac{(7 + 5 i) i}{- 7 i i}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

$\frac{7 i + 5 i i}{- 7 i i}$

Step 2.2.2

Multiply $5 i i$ .

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Step 2.2.2.1

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

$\frac{7 i + 5 (i^{1} i)}{- 7 i i}$

Step 2.2.2.2

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

$\frac{7 i + 5 (i^{1} i^{1})}{- 7 i i}$

Step 2.2.2.3

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

$\frac{7 i + 5 i^{1 + 1}}{- 7 i i}$

Step 2.2.2.4

Add $1$ and $1$ .

$\frac{7 i + 5 i^{2}}{- 7 i i}$

Step 2.2.3

Simplify each term.

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Step 2.2.3.1

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

$\frac{7 i + 5 \cdot - 1}{- 7 i i}$

Step 2.2.3.2

Multiply $5$ by $- 1$ .

$\frac{7 i - 5}{- 7 i i}$

Step 2.2.4

Reorder $7 i$ and $- 5$ .

$\frac{- 5 + 7 i}{- 7 i i}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Add parentheses.

$\frac{- 5 + 7 i}{- 7 (i i)}$

Step 2.3.2

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

$\frac{- 5 + 7 i}{- 7 (i^{1} i)}$

Step 2.3.3

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

$\frac{- 5 + 7 i}{- 7 (i^{1} i^{1})}$

Step 2.3.4

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

$\frac{- 5 + 7 i}{- 7 i^{1 + 1}}$

Step 2.3.5

Add $1$ and $1$ .

$\frac{- 5 + 7 i}{- 7 i^{2}}$

Step 2.3.6

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

$\frac{- 5 + 7 i}{- 7 \cdot - 1}$

Step 3

Multiply $- 7$ by $- 1$ .

$\frac{- 5 + 7 i}{7}$

Step 4

Split the fraction $\frac{- 5 + 7 i}{7}$ into two fractions.

$\frac{- 5}{7} + \frac{7 i}{7}$

Step 5

Simplify each term.

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Step 5.1

Move the negative in front of the fraction.

$- \frac{5}{7} + \frac{7 i}{7}$

Step 5.2

Cancel the common factor of $7$ .

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Step 5.2.1

Cancel the common factor.

$- \frac{5}{7} + \frac{7 i}{7}$

Step 5.2.2

Divide $i$ by $1$ .

$- \frac{5}{7} + i$