Using differentials, find the approximate value of each of the following up to 3 places of decimal.(i) `sqrt(25. 3)` (ii) `sqrt(49. 5)` (iii) `sqrt(0. 6)` (iv) `(0. 009)^(1/3)` (v) `(0. 999)^(1/(10))`(vi) `(15)^(1/4)`

Explanation

To find the approximate values using differentials for the given expressions, we will follow a systematic approach for each part of the question.

### (i) Approximate value of `sqrt(25.3)`

1. **Choose a nearby perfect square**: Let \( x = 25 \) (since \( \sqrt{25} = 5 \)).
2. **Calculate change in x**: \( \Delta x = 25.3 - 25 = 0.3 \).
3. **Define the function**: Let \( y = \sqrt{x} \).
4. **Find the derivative**:
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