## Simplifying the Expression: (ii) (4)^(-1)-(5)^(-1) ^(2)times((5)/(8))^(-1)

This expression involves negative exponents and fractions. Let's break it down step-by-step to simplify it:

### Understanding Negative Exponents

Remember that a negative exponent means taking the reciprocal of the base raised to the positive version of the exponent. For example:

- x⁻¹ = 1/x
- y⁻² = 1/y²

### Applying the Rules to the Expression

Let's apply this to our expression:

**(4)^(-1) = 1/4****(5)^(-1) = 1/5****(5/8)^(-1) = 8/5**(Reciprocal of the fraction)

Now our expression becomes:

**(1/4) - (1/5)² * (8/5)**

### Simplifying Further

**(1/5)² = 1/25****(1/4) - (1/25) * (8/5)**

Now we need to perform the multiplication:

**(1/4) - (8/125)**

### Finding a Common Denominator

To subtract fractions, they must have a common denominator. The least common multiple of 4 and 125 is 500.

**(125/500) - (32/500)**

### Final Calculation

Now we can subtract:

**(125 - 32) / 500 = 93/500**

Therefore, the simplified form of the expression **(ii) (4)^(-1)-(5)^(-1) ^(2)times((5)/(8))^(-1)** is **93/500**.