Find the value of :
`(i) ((5)/(8))^(-1)" "(ii)((-4)/(9))^(-1)" "(iii) (-7)^(-1) " " (iv)((1)/(-3))^(-1)`

Answer

To find the value of the given expressions, we will use the property of negative exponents, which states that \( a^{-n} = \frac{1}{a^n} \).

Let's solve each part step by step:

### (i) \( \left(\frac{5}{8}\right)^{-1} \)

1. **Apply the negative exponent rule**:
\[ \left(\frac{5}{8}\right)^{-1} = \frac{1}{\left(\frac{5}{8}\right)} \]

2. **Invert the fraction**:
\[ = \frac{8}{5} \]

**Final answer**: \( \frac{8}{5} \)

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### (ii) \( \left(-\frac{4}{9}\right)^{-1} \)

1. **Apply the negative exponent rule**:
\[ \left(-\frac{4}{9}\right)^{-1} = \frac{1}{\left(-\frac{4}{9}\right)} \]

2. **Invert the fraction**:
\[ = -\frac{9}{4} \]

**Final answer**: \( -\frac{9}{4} \)

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### (iii) \( (-7)^{-1} \)

1. **Apply the negative exponent rule**:
\[ (-7)^{-1} = \frac{1}{-7} \]

**Final answer**: \( -\frac{1}{7} \)

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### (iv) \( \left(\frac{1}{-3}\right)^{-1} \)

1. **Apply the negative exponent rule**:
\[ \left(\frac{1}{-3}\right)^{-1} = \frac{1}{\left(\frac{1}{-3}\right)} \]

2. **Invert the fraction**:
\[ = -3 \]

**Final answer**: \( -3 \)

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### Summary of Answers:
1. \( \left(\frac{5}{8}\right)^{-1} = \frac{8}{5} \)
2. \( \left(-\frac{4}{9}\right)^{-1} = -\frac{9}{4} \)
3. \( (-7)^{-1} = -\frac{1}{7} \)
4. \( \left(\frac{1}{-3}\right)^{-1} = -3 \)

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