Question

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

`3sqrt(243) + 6sqrt(72) - 3sqrt(8)`

Answer 1

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

Find the value of [ 3\sqrt{243} + 6\sqrt{72} - 3\sqrt{8} ]

Step-wise calculation:

1. Simplify the square roots by expressing the numbers under the root in terms of prime factors and perfect squares:

`sqrt(243) = sqrt(3^5)`

= `sqrt(3^4 xx 3)`

= `sqrt((3^2)^2 xx 3)`

= `9sqrt(3)`

`sqrt(72) = sqrt(36 xx 2)`

= `6sqrt(2)`

`sqrt(8) = sqrt(4 xx 2)`

= `2sqrt(2)`

2. Substitute these back into the expression:

`3sqrt(243) + 6sqrt(72) - 3sqrt(8)`

= `3 xx 9sqrt(3) + 6 xx 6sqrt(2) - 3 xx 2sqrt(2)`

= `27sqrt(3) + 36sqrt(2) - 6sqrt(2)`

3. Simplify the terms:

`27sqrt(3) + (36 - 6)sqrt(2)`

= `27sqrt(3) + 30sqrt(2)`

4. Substitute the approximate values for `sqrt(2)` and `sqrt(3)`:

= 27 × 1.732 + 30 × 1.414

= 46.764 + 42.42

= 89.184