Answer:
The exact value of x is 2.
Explanation:
To find the exact value of x, we need to solve the given equation for x. The equation is:
32^1/25 x 2^x = 8^1/4
To solve this equation, we need to rewrite it in a form that allows us to isolate the variable x. We can start by using the properties of exponents to simplify the left-hand side of the equation. We can use the property that says that raising a number to a fractional power is the same as taking the square root of that number, raised to the power of the denominator of the fraction. So, we can write:
(32^1/25)^2 = (32^1/25)^25/25 = 32^1/25
This simplification allows us to rewrite the left-hand side of the equation as follows:
32^1/25 * 2^x = (32^1/25)^2 * 2^x
Next, we can use the property of exponents that says that the product of two numbers raised to the same power is equal to the product of the numbers, raised to that power. So, we can rewrite the left-hand side of the equation as follows:
(32 * 2)^1/25 * 2^x = (32^1/25)^2 * 2^x
Then, we can use the property of exponents that says that raising a product to a power is the same as raising each factor to that power. So, we can rewrite the left-hand side of the equation as follows:
(32^1/25) * 2^(1/25 + x) = (32^1/25)^2 * 2^x
Finally, we can use the property of exponents that says that the exponent of a power is equal to the product of the exponents of the base and the power. So, we can rewrite the left-hand side of the equation as follows:
(32^1/25) * 2^(1/25 * 25 + 1 * x) = (32^1/25)^2 * 2^x
Since the right-hand side of the equation is already in a form that allows us to isolate the variable x, we can now use the property of exponents that says that raising a number to a power and then taking the square root of that number is the same as