write the square root of 32as a power of 2 and hence solve the equation 16to the power of x-1=2 square root of 32
since 32 = 2^5,
sqrt(32) = 32^(1/2) = (2^5)^(1/2) = 2^(5/2)
16^(x-1) = 2 * 2^(5/2) = 2^(1 + 5/2) = 2^(7/2)
16 = 2^4
2^(4x-4) = 2^(7/2)
so,
4x-4 = 7/2
8x-8 = 7
8x = 15
x = 15/8
To write the square root of 32 as a power of 2, we can express it as 2 raised to the power of (1/2).
Now let's solve the equation 16^x-1 = 2√32 by using the power properties of exponents.
First, let's simplify the equation by expressing 2√32 as a power of 2:
2√32 = 2√(2^5) = 2 * 2^(5/2) = 2^(1 + 5/2) = 2^(1 + 2.5) = 2^3.5
So, our equation becomes:
16^(x-1) = 2^3.5
Now, we can rewrite 16 and 2 as powers of 2:
(2^4)^(x-1) = 2^3.5
Using the power property of exponents, we multiply the exponents when raising a power to another power:
2^(4 * (x-1)) = 2^3.5
Since the bases are the same, the exponents must be equal:
4 * (x-1) = 3.5
Now, let's solve for x:
4x - 4 = 3.5
4x = 3.5 + 4
4x = 7.5
x = 7.5 / 4
x = 1.875
Therefore, the solution to the equation 16^(x-1) = 2√32 is x = 1.875.