Solve (e^(2x)-e^(-2x))^(2)

Question

Simplify the expression

e^{4 x} - 2 + e^{- 4 x}

Evaluate

(e^{2 x} - e^{- 2 x})^{2}

Solution

More Steps

Use the the distributive property to expand the expression

e^{2 x} \times e^{2 x} + e^{2 x} (- e^{- 2 x}) - e^{- 2 x} \times e^{2 x} - e^{- 2 x} (- e^{- 2 x})

Multiply the terms

More Steps

Evaluate

e^{2 x} \times e^{2 x}

Multiply the terms with the same base by adding their exponents

e^{2 x + 2 x}

Calculate

e^{4 x}

e^{4 x} + e^{2 x} (- e^{- 2 x}) - e^{- 2 x} \times e^{2 x} - e^{- 2 x} (- e^{- 2 x})

Multiply the terms

More Steps

Evaluate

e^{2 x} (- e^{- 2 x})

Multiply the terms with the same base by adding their exponents

- e^{2 x - 2 x}

Calculate

- e^{0}

Calculate

- 1

e^{4 x} - 1 - e^{- 2 x} \times e^{2 x} - e^{- 2 x} (- e^{- 2 x})

Multiply the terms

More Steps

Evaluate

- e^{- 2 x} \times e^{2 x}

Multiply the terms with the same base by adding their exponents

- e^{- 2 x + 2 x}

Calculate

- e^{0}

Calculate

- 1

e^{4 x} - 1 - 1 - e^{- 2 x} (- e^{- 2 x})

Multiply the terms

More Steps

Evaluate

- e^{- 2 x} (- e^{- 2 x})

Multiply the terms with the same base by adding their exponents

e^{- 2 x - 2 x}

Calculate

e^{- 4 x}

e^{4 x} - 1 - 1 + e^{- 4 x}

Calculate

e^{4 x} - 2 + e^{- 4 x}

e^{4 x} - 2 + e^{- 4 x}

Show Solution

x = 0

Evaluate

(e^{2 x} - e^{- 2 x})^{2}

To find the roots of the expression,set the expression equal to 0

(e^{2 x} - e^{- 2 x})^{2} = 0

The only way a power can be 0 is when the base equals 0

e^{2 x} - e^{- 2 x} = 0

Factor the expression

More Steps

Factor the expression

e^{2 x} - e^{- 2 x}

Evaluate

(e^{x})^{2} - (e^{x})^{- 2}

Rewrite the expression

(e^{x})^{- 2} (e^{x})^{4} - (e^{x})^{- 2}

Factor out (e^{x})^{- 2} from the expression

(e^{x})^{- 2} ((e^{x})^{4} - 1)

Use a^{2} - b^{2} = (a - b) (a + b) to factor the expression

(e^{x})^{- 2} ((e^{x})^{2} - 1) ((e^{x})^{2} + 1)

Use a^{2} - b^{2} = (a - b) (a + b) to factor the expression

(e^{x})^{- 2} (e^{x} - 1) (e^{x} + 1) ((e^{x})^{2} + 1)

Factor the expression

(e^{x} - 1) (e^{x} + 1) (e^{2 x} + 1) (e^{x})^{- 2}

(e^{x} - 1) (e^{x} + 1) (e^{2 x} + 1) (e^{x})^{- 2} = 0

Rewrite the expression

\frac{e ^{4 x} - 1}{( e ^{x} ) ^{2}} = 0

Cross multiply

e^{4 x} - 1 = (e^{x})^{2} \times 0

Simplify the equation

e^{4 x} - 1 = 0

Move the expression to the right side

e^{4 x} = 1

Write the number in exponential form with the base of e

e^{4 x} = e^{0}

Since the bases are the same,set the exponents equal

4 x = 0

Solution

x = 0

Show Solution