%5E2%2F3-divided-by-(81%2F16)%5E-1%2F4.jpeg "(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
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(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
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(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.