Let x^2/3 = 64, then find the value of x: (CSS MPT 2022)
A. 4
B. 8
C. 16
D. 512
To solve for x when (x^\frac{2}{3} = 64), you can follow these steps:
Step 1: Start with the equation
[x^\frac{2}{3} = 64]
Step 2: To eliminate the fractional exponent, you can raise both sides of the equation to the reciprocal of the fractional exponent, which is (3/2). This will give you:
[(x^\frac{2}{3})^\frac{3}{2} = 64^\frac{3}{2}]
Step 3: Simplify both sides. On the left side, the fractional exponents cancel out:
[x = 64^\frac{3}{2}]
Step 4: Calculate (64^\frac{3}{2}). To do this, first find the square root of 64, which is 8, and then raise that result to the power of 3:
[x = 8^3]
Step 5: Calculate the cube of 8:
[x = 8 \times 8 \times 8 = 512]
So, the value of (x) is 512.
To check your answer, you can substitute 512 back into the original equation to make sure it satisfies the equation:
[(512)^\frac{2}{3} = 64]
Simplify the left side:
[(8^3)^\frac{2}{3} = 64]
Again, the fractional exponents cancel out:
[8^{(3 \times \frac{2}{3})} = 64]
Simplify further:
[8^2 = 64]
Which is true. So, (x = 512) is the correct solution.
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