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Question
\left(e^{x}-e^{-x}\right)^{2}
Simplify the expression
e^{2x}-2+e^{-2x}
Evaluate
\left(e^{x}-e^{-x}\right)^{2}
Solution
More Steps
Use the the distributive property to expand the expression
e^{x}\times e^{x}+e^{x}\left(-e^{-x}\right)-e^{-x}\times e^{x}-e^{-x}\left(-e^{-x}\right)
Multiply the terms
More Steps
Evaluate
e^{x}\times e^{x}
Multiply the terms with the same base by adding their exponents
e^{x+x}
Calculate
e^{2x}
e^{2x}+e^{x}\left(-e^{-x}\right)-e^{-x}\times e^{x}-e^{-x}\left(-e^{-x}\right)
Multiply the terms
More Steps
Evaluate
e^{x}\left(-e^{-x}\right)
Multiply the terms with the same base by adding their exponents
-e^{x-x}
Calculate
-e^{0}
Calculate
-1
e^{2x}-1-e^{-x}\times e^{x}-e^{-x}\left(-e^{-x}\right)
Multiply the terms
More Steps
Evaluate
-e^{-x}\times e^{x}
Multiply the terms with the same base by adding their exponents
-e^{-x+x}
Calculate
-e^{0}
Calculate
-1
e^{2x}-1-1-e^{-x}\left(-e^{-x}\right)
Multiply the terms
More Steps
Evaluate
-e^{-x}\left(-e^{-x}\right)
Multiply the terms with the same base by adding their exponents
e^{-x-x}
Calculate
e^{-2x}
e^{2x}-1-1+e^{-2x}
Calculate
e^{2x}-2+e^{-2x}
e^{2x}-2+e^{-2x}
Show Solutions
Find the roots
x=0
Evaluate
\left(e^{x}-e^{-x}\right)^{2}
To find the roots of the expression,set the expression equal to 0
\left(e^{x}-e^{-x}\right)^{2}=0
The only way a power can be 0 is when the base equals 0
e^{x}-e^{-x}=0
Factor the expression
More Steps
Factor the expression
e^{x}-e^{-x}
Evaluate
e^{x}-\left(e^{x}\right)^{-1}
Rewrite the expression
\left(e^{x}\right)^{-1}\left(e^{x}\right)^{2}-\left(e^{x}\right)^{-1}
\text{Factor out }\left(e^{x}\right)^{-1}\text{ from the expression}
\left(e^{x}\right)^{-1}\left(\left(e^{x}\right)^{2}-1\right)
\text{Use }a^2-b^2=(a-b)(a+b)\text{ to factor the expression}
\left(e^{x}\right)^{-1}\left(e^{x}-1\right)\left(e^{x}+1\right)
Factor the expression
\left(e^{x}-1\right)\left(e^{x}+1\right)\left(e^{x}\right)^{-1}
\left(e^{x}-1\right)\left(e^{x}+1\right)\left(e^{x}\right)^{-1}=0
Rewrite the expression
\frac{e^{2x}-1}{e^{x}}=0
Cross multiply
e^{2x}-1=e^{x}\times 0
Simplify the equation
e^{2x}-1=0
Move the expression to the right side
e^{2x}=1
\text{Write the number in exponential form with the base of }e
e^{2x}=e^{0}
Since the bases are the same,set the exponents equal
2x=0
Solution
x=0
Show Solutions
