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Let's simplify each of the given expressions step by step.

### 1. \( 27^{\frac{2}{3}} \)
The fractional exponent \( \frac{2}{3} \) means we take the cube root of 27 and then square it.
- Cube root of 27 is \( 3 \), because \( 3^3 = 27 \).
- Now, square \( 3 \), so \( 3^2 = 9 \).

**Simplified result:** \( 27^{\frac{2}{3}} = 9 \).

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### 2. \( 64^{\frac{4}{3}} \)
Here, \( \frac{4}{3} \) means we take the cube root of 64 and then raise it to the 4th power.
- Cube root of 64 is \( 4 \), because \( 4^3 = 64 \).
- Now, raise \( 4 \) to the 4th power: \( 4^4 = 256 \).

**Simplified result:** \( 64^{\frac{4}{3}} = 256 \).

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### 3. \( 125^{-\frac{2}{3}} \)
The negative exponent means we take the reciprocal after performing the exponentiation. 
- Cube root of 125 is \( 5 \), because \( 5^3 = 125 \).
- Then, square \( 5 \), so \( 5^2 = 25 \).
- The negative exponent gives us the reciprocal: \( \frac{1}{25} \).

**Simplified result:** \( 125^{-\frac{2}{3}} = \frac{1}{25} \).

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### 4. \( \left( \frac{49}{121} \right)^{\frac{1}{2}} \)
The exponent \( \frac{1}{2} \) means we take the square root of the fraction.
- Square root of 49 is \( 7 \), and square root of 121 is \( 11 \).

**Simplified result:** \( \left( \frac{49}{121} \right)^{\frac{1}{2}} = \frac{7}{11} \).

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### 5. \( \left( \frac{16}{25} \right)^{-\frac{3}{2}} \)
The exponent \( -\frac{3}{2} \) involves two steps: taking the square root, cubing it, and then applying the negative to take the reciprocal.
- Square root of 16 is \( 4 \), and square root of 25 is \( 5 \), so \( \left( \frac{16}{25} \right)^{\frac{1}{2}} = \frac{4}{5} \).
- Now, cube it: \( \left( \frac{4}{5} \right)^3 = \frac{64}{125} \).
- Apply the negative exponent, which gives the reciprocal: \( \frac{125}{64} \).

**Simplified result:** \( \left( \frac{16}{25} \right)^{-\frac{3}{2}} = \frac{125}{64} \).

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Would you like more detailed explanations for any step or have any questions?

Here are 5 additional practice questions:
1. Simplify \( 32^{\frac{5}{2}} \).
2. Find \( \left( \frac{81}{16} \right)^{\frac{3}{4}} \).
3. Evaluate \( 144^{-\frac{1}{2}} \).
4. Simplify \( 243^{\frac{4}{5}} \).
5. Find \( \left( \frac{8}{27} \right)^{\frac{2}{3}} \).

**Tip:** When dealing with fractional exponents, always think of the denominator as the root and the numerator as the power.

Simplify the following expressions:
1. 27^(2/3)
2. 64^(4/3)
3. 125^(-2/3)
4. (49/121)^(1/2)
5. (16/25)^(-3/2)

Learn how to simplify fractional exponents such as 27^(2/3), 64^(4/3), and (49/121)^(1/2). Understand the laws of exponents and how to handle cube roots, square roots, and negative exponents in a step-by-step solution. Suitable for Grades 9-10 students.

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