(x^(5))^(1/3)(16x^(3))^(2/3)((1)/(4)x^(4/9))^(-3/2)=

2 min read Jun 17, 2024
(x^(5))^(1/3)(16x^(3))^(2/3)((1)/(4)x^(4/9))^(-3/2)=

Simplifying Expressions with Fractional Exponents

This article will guide you through simplifying the expression:

(x^(5))^(1/3)(16x^(3))^(2/3)((1)/(4)x^(4/9))^(-3/2)

Let's break it down step-by-step:

1. Applying Power of a Power Rule

We begin by applying the power of a power rule, which states: (a^m)^n = a^(m*n).

  • (x^(5))^(1/3) = x^(5/3)
  • (16x^(3))^(2/3) = 16^(2/3) * x^(2/3 * 3) = 4x^2
  • ((1)/(4)x^(4/9))^(-3/2) = (1/4)^(-3/2) * x^(4/9 * -3/2) = 8x^(-2/3)

2. Combining Terms

Now, we have:

x^(5/3) * 4x^2 * 8x^(-2/3)

To multiply these terms, we add the exponents of the 'x' terms:

x^(5/3 + 2 - 2/3) * 4 * 8

3. Simplifying the Exponent

Simplifying the exponent:

x^(5/3 + 6/3 - 2/3) * 32 = x^(9/3) * 32 = x^3 * 32

4. Final Result

Therefore, the simplified expression is:

32x^3

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