https://file.aiquickdraw.com/mathbot/file/17289842280993ybunss0.jpg

The expression in the image is \(\left( \frac{81}{256} \right)^{\frac{3}{4}}\).

Let's break this down and solve it step by step:

1. Start by recognizing that \(\frac{81}{256}\) can be written as the fraction of two perfect powers:
   - \(81 = 3^4\)
   - \(256 = 4^4\)

   So, \(\frac{81}{256} = \left(\frac{3}{4}\right)^4\).

2. Now apply the exponentiation to \(\left(\frac{3}{4}\right)^4\):
   \[
   \left( \frac{3}{4} \right)^4
   \]

3. Raise this fraction to the power \(\frac{3}{4}\):
   \[
   \left(\left(\frac{3}{4}\right)^4\right)^{\frac{3}{4}} = \left(\frac{3}{4}\right)^{4 \cdot \frac{3}{4}} = \left(\frac{3}{4}\right)^3
   \]

4. Now, evaluate \(\left(\frac{3}{4}\right)^3\):
   \[
   \left(\frac{3}{4}\right)^3 = \frac{3^3}{4^3} = \frac{27}{64}
   \]

So, the value of \(\left( \frac{81}{256} \right)^{\frac{3}{4}}\) is \(\frac{27}{64}\).

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you simplify expressions involving fractional exponents?
2. What is the significance of perfect powers in simplifying fractions?
3. How do you handle negative exponents in fractions?
4. What happens when the base of a fraction has different powers in the numerator and denominator?
5. Can fractional exponents also be applied to radicals (roots)?

**Tip:** To solve fractional exponents, remember that \(\frac{a}{b}\) as an exponent represents a combination of a root and a power.

Learn how to solve the expression (81/256)^(3/4) using fractional exponents. This detailed step-by-step solution covers key concepts in powers, roots, and exponent multiplication rules, suitable for students in Grades 9-11.

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