Trigonometry Examples

Simplify ( square root of 6(cos(120)+isin(120)))^4
Step 1
Simplify each term.
Tap for more steps...
Step 1.1
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.
Step 1.2
The exact value of is .
Step 1.3
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 1.4
The exact value of is .
Step 1.5
Combine and .
Step 2
Simplify terms.
Tap for more steps...
Step 2.1
Apply the distributive property.
Step 2.2
Combine and .
Step 3
Multiply .
Tap for more steps...
Step 3.1
Combine and .
Step 3.2
Combine using the product rule for radicals.
Step 3.3
Multiply by .
Step 4
Simplify the numerator.
Tap for more steps...
Step 4.1
Rewrite as .
Tap for more steps...
Step 4.1.1
Factor out of .
Step 4.1.2
Rewrite as .
Step 4.2
Pull terms out from under the radical.
Step 5
Use the Binomial Theorem.
Step 6
Simplify terms.
Tap for more steps...
Step 6.1
Simplify each term.
Tap for more steps...
Step 6.1.1
Use the power rule to distribute the exponent.
Tap for more steps...
Step 6.1.1.1
Apply the product rule to .
Step 6.1.1.2
Apply the product rule to .
Step 6.1.2
Raise to the power of .
Step 6.1.3
Multiply by .
Step 6.1.4
Simplify the numerator.
Tap for more steps...
Step 6.1.4.1
Rewrite as .
Tap for more steps...
Step 6.1.4.1.1
Use to rewrite as .
Step 6.1.4.1.2
Apply the power rule and multiply exponents, .
Step 6.1.4.1.3
Combine and .
Step 6.1.4.1.4
Cancel the common factor of and .
Tap for more steps...
Step 6.1.4.1.4.1
Factor out of .
Step 6.1.4.1.4.2
Cancel the common factors.
Tap for more steps...
Step 6.1.4.1.4.2.1
Factor out of .
Step 6.1.4.1.4.2.2
Cancel the common factor.
Step 6.1.4.1.4.2.3
Rewrite the expression.
Step 6.1.4.1.4.2.4
Divide by .
Step 6.1.4.2
Raise to the power of .
Step 6.1.5
Raise to the power of .
Step 6.1.6
Cancel the common factor of and .
Tap for more steps...
Step 6.1.6.1
Factor out of .
Step 6.1.6.2
Cancel the common factors.
Tap for more steps...
Step 6.1.6.2.1
Factor out of .
Step 6.1.6.2.2
Cancel the common factor.
Step 6.1.6.2.3
Rewrite the expression.
Step 6.1.7
Cancel the common factor of .
Tap for more steps...
Step 6.1.7.1
Factor out of .
Step 6.1.7.2
Cancel the common factor.
Step 6.1.7.3
Rewrite the expression.
Step 6.1.8
Multiply by .
Step 6.1.9
Use the power rule to distribute the exponent.
Tap for more steps...
Step 6.1.9.1
Apply the product rule to .
Step 6.1.9.2
Apply the product rule to .
Step 6.1.10
Raise to the power of .
Step 6.1.11
Simplify the numerator.
Tap for more steps...
Step 6.1.11.1
Rewrite as .
Step 6.1.11.2
Raise to the power of .
Step 6.1.11.3
Rewrite as .
Tap for more steps...
Step 6.1.11.3.1
Factor out of .
Step 6.1.11.3.2
Rewrite as .
Step 6.1.11.4
Pull terms out from under the radical.
Step 6.1.12
Raise to the power of .
Step 6.1.13
Cancel the common factor of .
Tap for more steps...
Step 6.1.13.1
Move the leading negative in into the numerator.
Step 6.1.13.2
Factor out of .
Step 6.1.13.3
Factor out of .
Step 6.1.13.4
Cancel the common factor.
Step 6.1.13.5
Rewrite the expression.
Step 6.1.14
Combine and .
Step 6.1.15
Multiply by .
Step 6.1.16
Cancel the common factor of and .
Tap for more steps...
Step 6.1.16.1
Factor out of .
Step 6.1.16.2
Cancel the common factors.
Tap for more steps...
Step 6.1.16.2.1
Factor out of .
Step 6.1.16.2.2
Cancel the common factor.
Step 6.1.16.2.3
Rewrite the expression.
Step 6.1.17
Move the negative in front of the fraction.
Step 6.1.18
Multiply .
Tap for more steps...
Step 6.1.18.1
Combine and .
Step 6.1.18.2
Combine using the product rule for radicals.
Step 6.1.18.3
Multiply by .
Step 6.1.18.4
Combine and .
Step 6.1.19
Simplify the numerator.
Tap for more steps...
Step 6.1.19.1
Rewrite as .
Tap for more steps...
Step 6.1.19.1.1
Factor out of .
Step 6.1.19.1.2
Rewrite as .
Step 6.1.19.2
Pull terms out from under the radical.
Step 6.1.19.3
Multiply by .
Step 6.1.20
Move to the left of .
Step 6.1.21
Cancel the common factor of and .
Tap for more steps...
Step 6.1.21.1
Factor out of .
Step 6.1.21.2
Cancel the common factors.
Tap for more steps...
Step 6.1.21.2.1
Factor out of .
Step 6.1.21.2.2
Cancel the common factor.
Step 6.1.21.2.3
Rewrite the expression.
Step 6.1.21.2.4
Divide by .
Step 6.1.22
Multiply by .
Step 6.1.23
Use the power rule to distribute the exponent.
Tap for more steps...
Step 6.1.23.1
Apply the product rule to .
Step 6.1.23.2
Apply the product rule to .
Step 6.1.24
Raise to the power of .
Step 6.1.25
Multiply by .
Step 6.1.26
Rewrite as .
Tap for more steps...
Step 6.1.26.1
Use to rewrite as .
Step 6.1.26.2
Apply the power rule and multiply exponents, .
Step 6.1.26.3
Combine and .
Step 6.1.26.4
Cancel the common factor of .
Tap for more steps...
Step 6.1.26.4.1
Cancel the common factor.
Step 6.1.26.4.2
Rewrite the expression.
Step 6.1.26.5
Evaluate the exponent.
Step 6.1.27
Raise to the power of .
Step 6.1.28
Cancel the common factor of .
Tap for more steps...
Step 6.1.28.1
Factor out of .
Step 6.1.28.2
Factor out of .
Step 6.1.28.3
Cancel the common factor.
Step 6.1.28.4
Rewrite the expression.
Step 6.1.29
Combine and .
Step 6.1.30
Multiply by .
Step 6.1.31
Divide by .
Step 6.1.32
Use the power rule to distribute the exponent.
Tap for more steps...
Step 6.1.32.1
Apply the product rule to .
Step 6.1.32.2
Apply the product rule to .
Step 6.1.32.3
Apply the product rule to .
Step 6.1.33
Simplify the numerator.
Tap for more steps...
Step 6.1.33.1
Raise to the power of .
Step 6.1.33.2
Rewrite as .
Tap for more steps...
Step 6.1.33.2.1
Use to rewrite as .
Step 6.1.33.2.2
Apply the power rule and multiply exponents, .
Step 6.1.33.2.3
Combine and .
Step 6.1.33.2.4
Cancel the common factor of .
Tap for more steps...
Step 6.1.33.2.4.1
Cancel the common factor.
Step 6.1.33.2.4.2
Rewrite the expression.
Step 6.1.33.2.5
Evaluate the exponent.
Step 6.1.33.3
Rewrite as .
Step 6.1.33.4
Combine exponents.
Tap for more steps...
Step 6.1.33.4.1
Multiply by .
Step 6.1.33.4.2
Multiply by .
Step 6.1.34
Raise to the power of .
Step 6.1.35
Cancel the common factor of and .
Tap for more steps...
Step 6.1.35.1
Factor out of .
Step 6.1.35.2
Cancel the common factors.
Tap for more steps...
Step 6.1.35.2.1
Factor out of .
Step 6.1.35.2.2
Cancel the common factor.
Step 6.1.35.2.3
Rewrite the expression.
Step 6.1.36
Move the negative in front of the fraction.
Step 6.1.37
Multiply .
Tap for more steps...
Step 6.1.37.1
Multiply by .
Step 6.1.37.2
Combine and .
Step 6.1.37.3
Multiply by .
Step 6.1.38
Move the negative in front of the fraction.
Step 6.1.39
Cancel the common factor of .
Tap for more steps...
Step 6.1.39.1
Move the leading negative in into the numerator.
Step 6.1.39.2
Factor out of .
Step 6.1.39.3
Cancel the common factor.
Step 6.1.39.4
Rewrite the expression.
Step 6.1.40
Multiply by .
Step 6.1.41
Use the power rule to distribute the exponent.
Tap for more steps...
Step 6.1.41.1
Apply the product rule to .
Step 6.1.41.2
Apply the product rule to .
Step 6.1.41.3
Apply the product rule to .
Step 6.1.42
Simplify the numerator.
Tap for more steps...
Step 6.1.42.1
Raise to the power of .
Step 6.1.42.2
Rewrite as .
Step 6.1.42.3
Raise to the power of .
Step 6.1.42.4
Rewrite as .
Tap for more steps...
Step 6.1.42.4.1
Factor out of .
Step 6.1.42.4.2
Rewrite as .
Step 6.1.42.5
Pull terms out from under the radical.
Step 6.1.42.6
Factor out .
Step 6.1.42.7
Rewrite as .
Step 6.1.42.8
Rewrite as .
Step 6.1.42.9
Combine exponents.
Tap for more steps...
Step 6.1.42.9.1
Factor out negative.
Step 6.1.42.9.2
Multiply by .
Step 6.1.42.9.3
Multiply by .
Step 6.1.43
Raise to the power of .
Step 6.1.44
Cancel the common factor of .
Tap for more steps...
Step 6.1.44.1
Factor out of .
Step 6.1.44.2
Factor out of .
Step 6.1.44.3
Cancel the common factor.
Step 6.1.44.4
Rewrite the expression.
Step 6.1.45
Combine and .
Step 6.1.46
Combine using the product rule for radicals.
Step 6.1.47
Multiply by .
Step 6.1.48
Cancel the common factor of and .
Tap for more steps...
Step 6.1.48.1
Factor out of .
Step 6.1.48.2
Cancel the common factors.
Tap for more steps...
Step 6.1.48.2.1
Factor out of .
Step 6.1.48.2.2
Cancel the common factor.
Step 6.1.48.2.3
Rewrite the expression.
Step 6.1.49
Simplify the numerator.
Tap for more steps...
Step 6.1.49.1
Rewrite as .
Tap for more steps...
Step 6.1.49.1.1
Factor out of .
Step 6.1.49.1.2
Rewrite as .
Step 6.1.49.2
Pull terms out from under the radical.
Step 6.1.49.3
Multiply by .
Step 6.1.50
Cancel the common factor of and .
Tap for more steps...
Step 6.1.50.1
Factor out of .
Step 6.1.50.2
Cancel the common factors.
Tap for more steps...
Step 6.1.50.2.1
Factor out of .
Step 6.1.50.2.2
Cancel the common factor.
Step 6.1.50.2.3
Rewrite the expression.
Step 6.1.50.2.4
Divide by .
Step 6.1.51
Multiply by .
Step 6.1.52
Use the power rule to distribute the exponent.
Tap for more steps...
Step 6.1.52.1
Apply the product rule to .
Step 6.1.52.2
Apply the product rule to .
Step 6.1.52.3
Apply the product rule to .
Step 6.1.53
Simplify the numerator.
Tap for more steps...
Step 6.1.53.1
Raise to the power of .
Step 6.1.53.2
Rewrite as .
Tap for more steps...
Step 6.1.53.2.1
Use to rewrite as .
Step 6.1.53.2.2
Apply the power rule and multiply exponents, .
Step 6.1.53.2.3
Combine and .
Step 6.1.53.2.4
Cancel the common factor of and .
Tap for more steps...
Step 6.1.53.2.4.1
Factor out of .
Step 6.1.53.2.4.2
Cancel the common factors.
Tap for more steps...
Step 6.1.53.2.4.2.1
Factor out of .
Step 6.1.53.2.4.2.2
Cancel the common factor.
Step 6.1.53.2.4.2.3
Rewrite the expression.
Step 6.1.53.2.4.2.4
Divide by .
Step 6.1.53.3
Raise to the power of .
Step 6.1.53.4
Rewrite as .
Tap for more steps...
Step 6.1.53.4.1
Rewrite as .
Step 6.1.53.4.2
Rewrite as .
Step 6.1.53.4.3
Raise to the power of .
Step 6.1.53.5
Combine exponents.
Tap for more steps...
Step 6.1.53.5.1
Multiply by .
Step 6.1.53.5.2
Multiply by .
Step 6.1.54
Raise to the power of .
Step 6.1.55
Cancel the common factor of and .
Tap for more steps...
Step 6.1.55.1
Factor out of .
Step 6.1.55.2
Cancel the common factors.
Tap for more steps...
Step 6.1.55.2.1
Factor out of .
Step 6.1.55.2.2
Cancel the common factor.
Step 6.1.55.2.3
Rewrite the expression.
Step 6.2
Combine fractions.
Tap for more steps...
Step 6.2.1
Combine the numerators over the common denominator.
Step 6.2.2
Add and .
Step 6.3
Simplify each term.
Tap for more steps...
Step 6.3.1
Cancel the common factor of and .
Tap for more steps...
Step 6.3.1.1
Factor out of .
Step 6.3.1.2
Cancel the common factors.
Tap for more steps...
Step 6.3.1.2.1
Factor out of .
Step 6.3.1.2.2
Cancel the common factor.
Step 6.3.1.2.3
Rewrite the expression.
Step 6.3.2
Move the negative in front of the fraction.
Step 6.4
Simplify terms.
Tap for more steps...
Step 6.4.1
Combine the numerators over the common denominator.
Step 6.4.2
Simplify the expression.
Tap for more steps...
Step 6.4.2.1
Subtract from .
Step 6.4.2.2
Divide by .
Step 6.4.2.3
Reorder the factors of .
Step 6.4.3
Add and .
Step 6.4.4
Reorder and .