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(b) (4)^(1/3)times(64)^(1/2)times(4)^(2/3)

2 min read Jun 16, 2024
(b) (4)^(1/3)times(64)^(1/2)times(4)^(2/3)

Simplifying the Expression: (b) (4)^(1/3) * (64)^(1/2) * (4)^(2/3)

This expression involves exponents and fractional powers. Let's break it down step-by-step to simplify it.

Understanding the Properties of Exponents

  • Fractional exponents represent roots: For example, x^(1/n) is the nth root of x. So, 4^(1/3) is the cube root of 4.
  • Product of powers: When multiplying exponents with the same base, add the powers. For example, x^m * x^n = x^(m+n).

Simplifying the Expression

  1. Simplify the square root: (64)^(1/2) is the square root of 64, which is 8.
  2. Combine the powers of 4: We have (4)^(1/3) * (4)^(2/3). Using the product of powers rule, we add the exponents: 1/3 + 2/3 = 1. Therefore, (4)^(1/3) * (4)^(2/3) = 4^1 = 4.

The Final Answer

Putting it all together, we have:

(4)^(1/3) * (64)^(1/2) * (4)^(2/3) = 4 * 8 * 4 = 128

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