Algebra Examples

$\frac{i \sqrt{2} - 5}{i + \sqrt{2}}$

Step 1

Multiply $\frac{i \sqrt{2} - 5}{i + \sqrt{2}}$ by $\frac{i - \sqrt{2}}{i - \sqrt{2}}$ .

$\frac{i \sqrt{2} - 5}{i + \sqrt{2}} \cdot \frac{i - \sqrt{2}}{i - \sqrt{2}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{i \sqrt{2} - 5}{i + \sqrt{2}}$ by $\frac{i - \sqrt{2}}{i - \sqrt{2}}$ .

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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

Step 3

Expand $(i \sqrt{2} - 5) (i - \sqrt{2})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $i \sqrt{2} i$ .

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Step 4.1.1.1

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

$\frac{i^{1} i \sqrt{2} + i \sqrt{2} (- \sqrt{2}) - 5 i - 5 (- \sqrt{2})}{- 3}$

Step 4.1.1.2

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

$\frac{i^{1} i^{1} \sqrt{2} + i \sqrt{2} (- \sqrt{2}) - 5 i - 5 (- \sqrt{2})}{- 3}$

Step 4.1.1.3

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

$\frac{i^{1 + 1} \sqrt{2} + i \sqrt{2} (- \sqrt{2}) - 5 i - 5 (- \sqrt{2})}{- 3}$

Step 4.1.1.4

Add $1$ and $1$ .

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

Step 4.1.2

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

$\frac{- 1 \sqrt{2} + i \sqrt{2} (- \sqrt{2}) - 5 i - 5 (- \sqrt{2})}{- 3}$

Step 4.1.3

Rewrite $- 1 \sqrt{2}$ as $- \sqrt{2}$ .

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

Step 4.1.4

Multiply $i \sqrt{2} (- \sqrt{2})$ .

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Step 4.1.4.1

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{- \sqrt{2} + i (- ({\sqrt{2}}^{1} \sqrt{2})) - 5 i - 5 (- \sqrt{2})}{- 3}$

Step 4.1.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{- \sqrt{2} + i (- ({\sqrt{2}}^{1} {\sqrt{2}}^{1})) - 5 i - 5 (- \sqrt{2})}{- 3}$

Step 4.1.4.3

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

$\frac{- \sqrt{2} + i (- {\sqrt{2}}^{1 + 1}) - 5 i - 5 (- \sqrt{2})}{- 3}$

Step 4.1.4.4

Add $1$ and $1$ .

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

Step 4.1.5

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 4.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{- \sqrt{2} + i (- {(2^{\frac{1}{2}})}^{2}) - 5 i - 5 (- \sqrt{2})}{- 3}$

Step 4.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{- \sqrt{2} + i (- 2^{\frac{1}{2} \cdot 2}) - 5 i - 5 (- \sqrt{2})}{- 3}$

Step 4.1.5.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 4.1.5.4

Cancel the common factor of $2$ .

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Step 4.1.5.4.1

Cancel the common factor.

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

Step 4.1.5.4.2

Rewrite the expression.

$\frac{- \sqrt{2} + i (- 2^{1}) - 5 i - 5 (- \sqrt{2})}{- 3}$

Step 4.1.5.5

Evaluate the exponent.

$\frac{- \sqrt{2} + i (- 1 \cdot 2) - 5 i - 5 (- \sqrt{2})}{- 3}$

Step 4.1.6

Multiply $- 1$ by $2$ .

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

Step 4.1.7

Move $- 2$ to the left of $i$ .

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

Step 4.1.8

Multiply $- 1$ by $- 5$ .

$\frac{- \sqrt{2} - 2 i - 5 i + 5 \sqrt{2}}{- 3}$

Step 4.2

Add $- \sqrt{2}$ and $5 \sqrt{2}$ .

$\frac{- 2 i - 5 i + 4 \sqrt{2}}{- 3}$

Step 4.3

Subtract $5 i$ from $- 2 i$ .

$\frac{- 7 i + 4 \sqrt{2}}{- 3}$

Step 5

Reorder $- 7 i$ and $4 \sqrt{2}$ .

$\frac{4 \sqrt{2} - 7 i}{- 3}$

Step 6

Move the negative in front of the fraction.

$- \frac{4 \sqrt{2} - 7 i}{3}$