Write the reciprocal of:
( i ) `(frac(2)(3))^(4)`
( ii ) `(frac(-3)(5))^(61)`
( iii ) `2^(5)`
( iv ) `(-5)^(6)`

Answer

To find the reciprocal of each expression, we need to follow the rule that the reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \). For a number raised to a power, we can treat it as a fraction with 1 in the denominator.

Let's solve each part step by step:

### (i) Reciprocal of \( \left(\frac{2}{3}\right)^{4} \)

1. **Identify the expression**: The expression is \( \left(\frac{2}{3}\right)^{4} \).
2. **Write the reciprocal**: The reciprocal will switch the numerator and denominator. Thus, it becomes \( \left(\frac{3}{2}\right)^{4} \).

**Final Answer**: \( \left(\frac{3}{2}\right)^{4} \)

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### (ii) Reciprocal of \( \left(\frac{-3}{5}\right)^{61} \)

1. **Identify the expression**: The expression is \( \left(\frac{-3}{5}\right)^{61} \).
2. **Write the reciprocal**: The reciprocal will switch the numerator and denominator. Thus, it becomes \( \left(\frac{5}{-3}\right)^{61} \).

**Final Answer**: \( \left(\frac{5}{-3}\right)^{61} \) or \( -\left(\frac{5}{3}\right)^{61} \)

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### (iii) Reciprocal of \( 2^{5} \)

1. **Identify the expression**: The expression is \( 2^{5} \).
2. **Rewrite as a fraction**: We can write this as \( \frac{2}{1}^{5} \).
3. **Write the reciprocal**: The reciprocal will switch the numerator and denominator. Thus, it becomes \( \left(\frac{1}{2}\right)^{5} \).

**Final Answer**: \( \left(\frac{1}{2}\right)^{5} \)

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### (iv) Reciprocal of \( (-5)^{6} \)

1. **Identify the expression**: The expression is \( (-5)^{6} \).
2. **Rewrite as a fraction**: We can write this as \( \frac{-5}{1}^{6} \).
3. **Write the reciprocal**: The reciprocal will switch the numerator and denominator. Thus, it becomes \( \left(\frac{1}{-5}\right)^{6} \).

**Final Answer**: \( \left(\frac{1}{-5}\right)^{6} \) or \( -\left(\frac{1}{5}\right)^{6} \)

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