Simplify: (2^(1+sqrt2)/2^(1-sqrt2))^sqrt2
I need all of the steps.
I need this ASAP because this assignment is due tomorrow.
Thanks
To simplify the expression (2^(1+sqrt2)/2^(1-sqrt2))^sqrt2, we can follow these steps:
Step 1: Simplify the exponent inside the parentheses:
- For the numerator, distribute the exponent (1+sqrt2) to each term:
2^(1+sqrt2) = 2^1 * 2^(sqrt2)
= 2 * 2^(sqrt2)
- For the denominator, do the same with (1-sqrt2):
2^(1-sqrt2) = 2^1 * 2^(-sqrt2)
= 2 * 2^(-sqrt2)
Step 2: Simplify the overall expression by dividing the numerator by the denominator:
(2^(1+sqrt2)/2^(1-sqrt2))^sqrt2 = (2 * 2^(sqrt2)) / (2 * 2^(-sqrt2))
Step 3: Cancel out the common factors in the numerator and denominator:
- In the numerator, divide 2 by 2: 2/2 = 1
- In the denominator, we can subtract the exponents: 2^(-sqrt2) = 1/2^(sqrt2)
Step 4: Simplify the expression further:
(2 * 2^(sqrt2)) / (2 * 2^(-sqrt2)) = (2^(sqrt2)) / (1/2^(sqrt2))
= 2^(sqrt2) * 2^(sqrt2)
= 2^(sqrt2 + sqrt2)
= 2^(2sqrt2)
= (2^2)^(sqrt2)
= 4^(sqrt2)
So, the simplified form of the expression (2^(1+sqrt2)/2^(1-sqrt2))^sqrt2 is 4^(sqrt2).
Make sure to double-check the steps and the final result to ensure accuracy.