Basic Math Examples

$\frac{\sqrt{3} (\cos (315) + i \sin (315))}{\sqrt{6} (\cos (45) + i \sin (45))}$

Step 1

Combine

$\sqrt{3}$ and

$\sqrt{6}$ into a single radical.

$\frac{\sqrt{\frac{3}{6}} (\cos (315) + i \sin (315))}{\cos (45) + i \sin (45)}$

Step 2

Cancel the common factor of

$3$ and

$6$ .

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Step 2.1

Factor

$3$ out of

$3$ .

$\frac{\sqrt{\frac{3 (1)}{6}} (\cos (315) + i \sin (315))}{\cos (45) + i \sin (45)}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor

$3$ out of

$6$ .

$\frac{\sqrt{\frac{3 \cdot 1}{3 \cdot 2}} (\cos (315) + i \sin (315))}{\cos (45) + i \sin (45)}$

Step 2.2.2

Cancel the common factor.

$\frac{\sqrt{\frac{3 \cdot 1}{3 \cdot 2}} (\cos (315) + i \sin (315))}{\cos (45) + i \sin (45)}$

Step 2.2.3

Rewrite the expression.

$\frac{\sqrt{\frac{1}{2}} (\cos (315) + i \sin (315))}{\cos (45) + i \sin (45)}$

Step 3

Simplify the numerator.

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Step 3.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{\sqrt{\frac{1}{2}} (\cos (45) + i \sin (315))}{\cos (45) + i \sin (45)}$

Step 3.2

The exact value of

$\cos (45)$ is

$\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{\frac{1}{2}} (\frac{\sqrt{2}}{2} + i \sin (315))}{\cos (45) + i \sin (45)}$

Step 3.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$\frac{\sqrt{\frac{1}{2}} (\frac{\sqrt{2}}{2} + i (- \sin (45)))}{\cos (45) + i \sin (45)}$

Step 3.4

The exact value of

$\sin (45)$ is

$\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{\frac{1}{2}} (\frac{\sqrt{2}}{2} + i (- \frac{\sqrt{2}}{2}))}{\cos (45) + i \sin (45)}$

Step 3.5

Combine

$i$ and

$\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{\frac{1}{2}} (\frac{\sqrt{2}}{2} - \frac{i \sqrt{2}}{2})}{\cos (45) + i \sin (45)}$

Step 3.6

Rewrite

$\sqrt{\frac{1}{2}}$ as

$\frac{\sqrt{1}}{\sqrt{2}}$ .

$\frac{\frac{\sqrt{1}}{\sqrt{2}} (\frac{\sqrt{2}}{2} - \frac{i \sqrt{2}}{2})}{\cos (45) + i \sin (45)}$

Step 3.7

Any root of

$1$ is

$1$ .

$\frac{\frac{1}{\sqrt{2}} (\frac{\sqrt{2}}{2} - \frac{i \sqrt{2}}{2})}{\cos (45) + i \sin (45)}$

Step 3.8

Multiply

$\frac{1}{\sqrt{2}}$ by

$\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} (\frac{\sqrt{2}}{2} - \frac{i \sqrt{2}}{2})}{\cos (45) + i \sin (45)}$

Step 3.9

Combine and simplify the denominator.

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Step 3.9.1

Multiply

$\frac{1}{\sqrt{2}}$ by

$\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\frac{\sqrt{2}}{\sqrt{2} \sqrt{2}} (\frac{\sqrt{2}}{2} - \frac{i \sqrt{2}}{2})}{\cos (45) + i \sin (45)}$

Step 3.9.2

Raise

$\sqrt{2}$ to the power of

$1$ .

$\frac{\frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} (\frac{\sqrt{2}}{2} - \frac{i \sqrt{2}}{2})}{\cos (45) + i \sin (45)}$

Step 3.9.3

Raise

$\sqrt{2}$ to the power of

$1$ .

$\frac{\frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} (\frac{\sqrt{2}}{2} - \frac{i \sqrt{2}}{2})}{\cos (45) + i \sin (45)}$

Step 3.9.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}} (\frac{\sqrt{2}}{2} - \frac{i \sqrt{2}}{2})}{\cos (45) + i \sin (45)}$

Step 3.9.5

Add

$1$ and

$1$ .

$\frac{\frac{\sqrt{2}}{{\sqrt{2}}^{2}} (\frac{\sqrt{2}}{2} - \frac{i \sqrt{2}}{2})}{\cos (45) + i \sin (45)}$

Step 3.9.6

Rewrite

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

$2$ .

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Step 3.9.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

$2^{\frac{1}{2}}$ .

$\frac{\frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} (\frac{\sqrt{2}}{2} - \frac{i \sqrt{2}}{2})}{\cos (45) + i \sin (45)}$

Step 3.9.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}} (\frac{\sqrt{2}}{2} - \frac{i \sqrt{2}}{2})}{\cos (45) + i \sin (45)}$

Step 3.9.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$\frac{\frac{\sqrt{2}}{2^{\frac{2}{2}}} (\frac{\sqrt{2}}{2} - \frac{i \sqrt{2}}{2})}{\cos (45) + i \sin (45)}$

Step 3.9.6.4

Cancel the common factor of

$2$ .

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Step 3.9.6.4.1

Cancel the common factor.

$\frac{\frac{\sqrt{2}}{2^{\frac{2}{2}}} (\frac{\sqrt{2}}{2} - \frac{i \sqrt{2}}{2})}{\cos (45) + i \sin (45)}$

Step 3.9.6.4.2

Rewrite the expression.

$\frac{\frac{\sqrt{2}}{2^{1}} (\frac{\sqrt{2}}{2} - \frac{i \sqrt{2}}{2})}{\cos (45) + i \sin (45)}$

Step 3.9.6.5

Evaluate the exponent.

$\frac{\frac{\sqrt{2}}{2} (\frac{\sqrt{2}}{2} - \frac{i \sqrt{2}}{2})}{\cos (45) + i \sin (45)}$

Step 3.10

Combine the numerators over the common denominator.

$\frac{\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2} - i \sqrt{2}}{2}}{\cos (45) + i \sin (45)}$

Step 4

Simplify the denominator.

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Step 4.1

The exact value of

$\cos (45)$ is

$\frac{\sqrt{2}}{2}$ .

$\frac{\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2} - i \sqrt{2}}{2}}{\frac{\sqrt{2}}{2} + i \sin (45)}$

Step 4.2

The exact value of

$\sin (45)$ is

$\frac{\sqrt{2}}{2}$ .

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

Step 4.3

Combine

$i$ and

$\frac{\sqrt{2}}{2}$ .

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

Step 4.4

Combine the numerators over the common denominator.

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

Step 5

Combine fractions.

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Step 5.1

Multiply

$\frac{\sqrt{2}}{2}$ by

$\frac{\sqrt{2} - i \sqrt{2}}{2}$ .

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

Step 5.2

Multiply

$2$ by

$2$ .

$\frac{\frac{\sqrt{2} (\sqrt{2} - i \sqrt{2})}{4}}{\frac{\sqrt{2} + i \sqrt{2}}{2}}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{\frac{\sqrt{2} \sqrt{2} + \sqrt{2} (- i \sqrt{2})}{4}}{\frac{\sqrt{2} + i \sqrt{2}}{2}}$

Step 6.2

Combine using the product rule for radicals.

$\frac{\frac{\sqrt{2 \cdot 2} + \sqrt{2} (- i \sqrt{2})}{4}}{\frac{\sqrt{2} + i \sqrt{2}}{2}}$

Step 6.3

Multiply

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

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Step 6.3.1

Raise

$\sqrt{2}$ to the power of

$1$ .

$\frac{\frac{\sqrt{2 \cdot 2} - i ({\sqrt{2}}^{1} \sqrt{2})}{4}}{\frac{\sqrt{2} + i \sqrt{2}}{2}}$

Step 6.3.2

Raise

$\sqrt{2}$ to the power of

$1$ .

$\frac{\frac{\sqrt{2 \cdot 2} - i ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{4}}{\frac{\sqrt{2} + i \sqrt{2}}{2}}$

Step 6.3.3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{\sqrt{2 \cdot 2} - i {\sqrt{2}}^{1 + 1}}{4}}{\frac{\sqrt{2} + i \sqrt{2}}{2}}$

Step 6.3.4

Add

$1$ and

$1$ .

$\frac{\frac{\sqrt{2 \cdot 2} - i {\sqrt{2}}^{2}}{4}}{\frac{\sqrt{2} + i \sqrt{2}}{2}}$

Step 6.4

Simplify each term.

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Step 6.4.1

Multiply

$2$ by

$2$ .

$\frac{\frac{\sqrt{4} - i {\sqrt{2}}^{2}}{4}}{\frac{\sqrt{2} + i \sqrt{2}}{2}}$

Step 6.4.2

Rewrite

$4$ as

$2^{2}$ .

$\frac{\frac{\sqrt{2^{2}} - i {\sqrt{2}}^{2}}{4}}{\frac{\sqrt{2} + i \sqrt{2}}{2}}$

Step 6.4.3

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\frac{2 - i {\sqrt{2}}^{2}}{4}}{\frac{\sqrt{2} + i \sqrt{2}}{2}}$

Step 6.4.4

Rewrite

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

$2$ .

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Step 6.4.4.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

$2^{\frac{1}{2}}$ .

$\frac{\frac{2 - i {(2^{\frac{1}{2}})}^{2}}{4}}{\frac{\sqrt{2} + i \sqrt{2}}{2}}$

Step 6.4.4.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{2 - i \cdot 2^{\frac{1}{2} \cdot 2}}{4}}{\frac{\sqrt{2} + i \sqrt{2}}{2}}$

Step 6.4.4.3

Combine

$\frac{1}{2}$ and

$2$ .

$\frac{\frac{2 - i \cdot 2^{\frac{2}{2}}}{4}}{\frac{\sqrt{2} + i \sqrt{2}}{2}}$

Step 6.4.4.4

Cancel the common factor of

$2$ .

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Step 6.4.4.4.1

Cancel the common factor.

$\frac{\frac{2 - i \cdot 2^{\frac{2}{2}}}{4}}{\frac{\sqrt{2} + i \sqrt{2}}{2}}$

Step 6.4.4.4.2

Rewrite the expression.

$\frac{\frac{2 - i \cdot 2^{1}}{4}}{\frac{\sqrt{2} + i \sqrt{2}}{2}}$

Step 6.4.4.5

Evaluate the exponent.

$\frac{\frac{2 - i \cdot 2}{4}}{\frac{\sqrt{2} + i \sqrt{2}}{2}}$

Step 6.4.5

Multiply

$2$ by

$- 1$ .

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

Step 6.5

Cancel the common factor of

$2 - 2 i$ and

$4$ .

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Step 6.5.1

Factor

$2$ out of

$2$ .

$\frac{\frac{2 (1) - 2 i}{4}}{\frac{\sqrt{2} + i \sqrt{2}}{2}}$

Step 6.5.2

Factor

$2$ out of

$- 2 i$ .

$\frac{\frac{2 (1) + 2 (- i)}{4}}{\frac{\sqrt{2} + i \sqrt{2}}{2}}$

Step 6.5.3

Factor

$2$ out of

$2 (1) + 2 (- i)$ .

$\frac{\frac{2 (1 - i)}{4}}{\frac{\sqrt{2} + i \sqrt{2}}{2}}$

Step 6.5.4

Cancel the common factors.

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Step 6.5.4.1

Factor

$2$ out of

$4$ .

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

Step 6.5.4.2

Cancel the common factor.

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

Step 6.5.4.3

Rewrite the expression.

$\frac{\frac{1 - i}{2}}{\frac{\sqrt{2} + i \sqrt{2}}{2}}$

Step 7

Multiply the numerator by the reciprocal of the denominator.

$\frac{1 - i}{2} \cdot \frac{2}{\sqrt{2} + i \sqrt{2}}$

Step 8

Cancel the common factor of

$2$ .

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Step 8.1

Cancel the common factor.

$\frac{1 - i}{2} \cdot \frac{2}{\sqrt{2} + i \sqrt{2}}$

Step 8.2

Rewrite the expression.

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

Step 9

Multiply

$\frac{1}{\sqrt{2} + i \sqrt{2}}$ by

$\frac{\sqrt{2} - i \sqrt{2}}{\sqrt{2} - i \sqrt{2}}$ .

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

Step 10

Simplify terms.

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Step 10.1

Multiply

$\frac{1}{\sqrt{2} + i \sqrt{2}}$ by

$\frac{\sqrt{2} - i \sqrt{2}}{\sqrt{2} - i \sqrt{2}}$ .

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

Step 10.2

Expand the denominator using the FOIL method.

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

Step 10.3

Simplify.

$(1 - i) \frac{\sqrt{2} - i \sqrt{2}}{4}$

Step 10.4

Apply the distributive property.

$1 \frac{\sqrt{2} - i \sqrt{2}}{4} - i \frac{\sqrt{2} - i \sqrt{2}}{4}$

Step 10.5

Multiply

$\frac{\sqrt{2} - i \sqrt{2}}{4}$ by

$1$ .

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

Step 10.6

Combine

$\frac{\sqrt{2} - i \sqrt{2}}{4}$ and

$i$ .

$\frac{\sqrt{2} - i \sqrt{2}}{4} - \frac{(\sqrt{2} - i \sqrt{2}) i}{4}$

Step 10.7

Combine the numerators over the common denominator.

$\frac{\sqrt{2} - i \sqrt{2} - (\sqrt{2} - i \sqrt{2}) i}{4}$

Step 11

Simplify each term.

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Step 11.1

Apply the distributive property.

$\frac{\sqrt{2} - i \sqrt{2} + (- \sqrt{2} - (- i \sqrt{2})) i}{4}$

Step 11.2

Multiply

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

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Step 11.2.1

Multiply

$- 1$ by

$- 1$ .

$\frac{\sqrt{2} - i \sqrt{2} + (- \sqrt{2} + 1 (i \sqrt{2})) i}{4}$

Step 11.2.2

Multiply

$\sqrt{2}$ by

$1$ .

$\frac{\sqrt{2} - i \sqrt{2} + (- \sqrt{2} + \sqrt{2} i) i}{4}$

Step 11.3

Apply the distributive property.

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

Step 11.4

Multiply

$\sqrt{2} i i$ .

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Step 11.4.1

Raise

$i$ to the power of

$1$ .

$\frac{\sqrt{2} - i \sqrt{2} - \sqrt{2} i + \sqrt{2} (i^{1} i)}{4}$

Step 11.4.2

Raise

$i$ to the power of

$1$ .

$\frac{\sqrt{2} - i \sqrt{2} - \sqrt{2} i + \sqrt{2} (i^{1} i^{1})}{4}$

Step 11.4.3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{2} - i \sqrt{2} - \sqrt{2} i + \sqrt{2} i^{1 + 1}}{4}$

Step 11.4.4

Add

$1$ and

$1$ .

$\frac{\sqrt{2} - i \sqrt{2} - \sqrt{2} i + \sqrt{2} i^{2}}{4}$

Step 11.5

Simplify each term.

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Step 11.5.1

Rewrite

$i^{2}$ as

$- 1$ .

$\frac{\sqrt{2} - i \sqrt{2} - \sqrt{2} i + \sqrt{2} \cdot - 1}{4}$

Step 11.5.2

Move

$- 1$ to the left of

$\sqrt{2}$ .

$\frac{\sqrt{2} - i \sqrt{2} - \sqrt{2} i - 1 \cdot \sqrt{2}}{4}$

Step 11.5.3

Rewrite

$- 1 \sqrt{2}$ as

$- \sqrt{2}$ .

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

Step 12

Simplify terms.

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Step 12.1

Subtract

$\sqrt{2}$ from

$\sqrt{2}$ .

$\frac{- i \sqrt{2} - \sqrt{2} i + 0}{4}$

Step 12.2

Reorder the factors of

$- \sqrt{2} i$ .

$\frac{- i \sqrt{2} - i \sqrt{2} + 0}{4}$

Step 12.3

Subtract

$i \sqrt{2}$ from

$- i \sqrt{2}$ .

$\frac{- 2 i \sqrt{2} + 0}{4}$

Step 12.4

Add

$- 2 i \sqrt{2}$ and

$0$ .

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

Step 12.5

Cancel the common factor of

$- 2$ and

$4$ .

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Step 12.5.1

Factor

$2$ out of

$- 2 i \sqrt{2}$ .

$\frac{2 (- i \sqrt{2})}{4}$

Step 12.5.2

Cancel the common factors.

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Step 12.5.2.1

Factor

$2$ out of

$4$ .

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

Step 12.5.2.2

Cancel the common factor.

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

Step 12.5.2.3

Rewrite the expression.

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

Step 12.6

Move the negative in front of the fraction.

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