Answer:
Step-by-step explanation:
To find the value of x in the equation (4^2)^(-1/2) = 64^(1/4)/(4^x+2), we can simplify each side of the equation step by step.
First, let's simplify the left side of the equation:
(4^2)^(-1/2) = 16^(-1/2)
Since (a^m)^n is equal to a^(m*n), we can simplify further:
16^(-1/2) = 1/16^(1/2)
The square root of 16 is 4, so:
1/16^(1/2) = 1/4
Now, let's simplify the right side of the equation:
64^(1/4) = (4^3)^(1/4)
Using the property of exponentiation, (a^m)^n = a^(m*n), we have:
(4^3)^(1/4) = 4^(3*(1/4))
Simplifying further:
4^(3*(1/4)) = 4^(3/4)
Now, let's simplify the denominator, 4^x+2:
4^x+2 = 4^(x+2)
Now, we can rewrite the equation as:
1/4 = 4^(3/4)/(4^(x+2))
To compare the sides of the equation, we can rewrite 1/4 as 4^(-1):
4^(-1) = 4^(3/4)/(4^(x+2))
Using the property of exponentiation that a^(-n) = 1/a^n, we have:
4^(3/4)/(4^(x+2)) = 4^(-1)
Now, to equate the bases, we can cancel out the common base, which is 4:
3/4 = -1 - (x+2)
Simplifying further, we have:
3/4 = -x - 3
Now, let's isolate the variable x by moving the constant terms to one side of the equation:
x = -3 - 3/4
To simplify the right side, we can convert 3 to a fraction with a denominator of 4:
x = -12/4 - 3/4
Combining the terms:
x = (-12 - 3)/4
x = -15/4
Therefore, the value of x is -15/4.
