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(27)^2/3 Divided By (81/16)^-1/4

2 min read Jun 16, 2024
(27)^2/3 Divided By (81/16)^-1/4

Simplifying the Expression: (27)^(2/3) divided by (81/16)^(-1/4)

This problem involves simplifying an expression with fractional exponents. Let's break down the steps:

Understanding Fractional Exponents

Fractional exponents represent roots and powers. For example, x^(m/n) means the n-th root of x raised to the power of m.

Step 1: Simplifying the terms individually

  • (27)^(2/3)

    • The cube root of 27 is 3 (since 3 x 3 x 3 = 27)
    • 3 squared is 9
    • Therefore, (27)^(2/3) = 9

  • (81/16)^(-1/4)

    • A negative exponent indicates the reciprocal. So, (81/16)^(-1/4) = (16/81)^(1/4)
    • The fourth root of 16 is 2 (since 2 x 2 x 2 x 2 = 16)
    • The fourth root of 81 is 3 (since 3 x 3 x 3 x 3 = 81)
    • Therefore, (16/81)^(1/4) = 2/3

Step 2: Dividing the Simplified Terms

Now we have: 9 divided by 2/3

  • Dividing by a fraction is the same as multiplying by its reciprocal.
  • The reciprocal of 2/3 is 3/2
  • Therefore, 9 divided by 2/3 is the same as 9 x (3/2) = 27/2

Final Answer

The simplified form of (27)^(2/3) divided by (81/16)^(-1/4) is 27/2.

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