Question

Solve the equation: x * sqrt[(x^1/2) * sqrt (x^1/4)] =1024

A) x=2
B) x=4
C) x=8
D) x=16

Answer 1

The equation x * sqrt[(x^1/2) * sqrt (x^1/4)] = 1024 simplifies to x^(7/4) = 1024. Taking the fourth root and then raising both sides to the power of 16/7 yields x=4. The correct answer is B) x=4.

Step-by-step explanation:

To solve the equation x * sqrt[(x^1/2) * sqrt (x^1/4)] = 1024, we first need to simplify the expression under the square root. We utilize the properties of exponents to combine the roots:

1. x^1/2 is the square root of x.
2. sqrt(x^1/4) is the fourth root of x, which can also be written as x^1/4.
3. Multiplying these together, we have x^1/2 * x^1/4, which simplifies to x^3/4 when we add the exponents.
4. So, the original equation becomes x * (x^3/4), which simplifies to x^(7/4).
5. To solve for x, we equate x^(7/4) = 1024, and take the fourth root of both sides to obtain x^(7/16) = 2, as 1024 is 2 to the power of 10.
6. Then, we raise both sides to the reciprocal of 7/16, which is 16/7, to isolate x. This gives us x = 2^(16/7).
7. Since 2^(16/7) simplifies to 2^2 or 4, the solution is x=4.

The correct answer is B) x=4.