Question

If `sqrt(2) = 1.414, sqrt(3) = 1.732`, find the value of the following:

`5sqrt(12) + sqrt(98) - sqrt(108)`

Answer 1

Given: `sqrt(2) = 1.414, sqrt(3) = 1.732`

Find the value of `5sqrt(12) + sqrt(98) - sqrt(108)`

Step-wise calculation:

1. Simplify the square roots inside the expression using prime factorization:

`sqrt(12) = sqrt(4 xx 3) = 2sqrt(3)`

`sqrt(98) = sqrt(49 xx 2) = 7sqrt(2)`

`sqrt(108) = sqrt(36 xx 3) = 6sqrt(3)`

2. Substitute these back into the expression:

`5sqrt(12) + sqrt(98) - sqrt(108)`

= `5 xx 2sqrt(3) + 7sqrt(2) - 6sqrt(3)` 

= `10sqrt(3) + 7sqrt(2) - 6sqrt(3)`

3. Combine like terms:

`(10sqrt(3) - 6sqrt(3)) + 7sqrt(2)`

= `4sqrt(3) + 7sqrt(2)`

4. Now, substitute the approximate values for `sqrt(2)` and `sqrt(3)`: 

4 × 1.732 + 7 × 1.414 

= 6.928 + 9.898

= 16.826