(A)Simplify:
n 1 n-1
1. 2 . 25
---------------
n
50.50
2. n 4 n 1
2 - 6.2
---------------
n 2
2 .5
(B)solve for x
(1 over 2) to the power x 1 =32
(A) To simplify these expressions, we need to apply the appropriate mathematical operations and combine like terms.
1. Simplifying the first expression:
We can rewrite the expression as follows:
n n-1
2 / 25
--------------
n
50.50
To simplify, we combine the terms with the same base (n) by performing subtraction:
2^(n) / 25^(n-1)
-------------------
n^(50.50)
2. Simplifying the second expression:
Similarly, we can rewrite the expression as:
n^4 n^1
2 - 6.2
--------------
n^2
2^0.5
To simplify, we can subtract the terms with the same base (n) and divide the exponents:
(2^(n^4) - 6.2) / (n^(2^0.5))
(B) To solve for x in the equation (1/2)^x = 32, we can use logarithms.
(1/2)^x = 32
To get rid of the exponent on the base (1/2), we can take the logarithm of both sides. Let's use the natural logarithm (ln) for this example:
ln((1/2)^x) = ln(32)
The logarithm property states that the exponent can be brought down as a coefficient:
x * ln(1/2) = ln(32)
Now, we can solve for x by dividing both sides by ln(1/2):
x = ln(32) / ln(1/2)
Using a calculator, we can evaluate this expression:
x ≈ -4.96
Therefore, the solution for x is approximately -4.96.