Question

How do you simplify the square root of 458?

Answer 1

This can be solved easily using a Calculator . We get the answer as `color(green)(approx 21.400935`

But if we are expected to do it without a calculator , here's how it can be done.

`sqrt458 = sqrt(2*229) = sqrt 2 * sqrt 229`

We know that `sqrt 2 approx 1.4142`

To find the `color(blue)(sqrt229`, we can use the following formula:

`sqrt(Non-Perfect-Square) approx sqrt(Nearest-Perfect-Square) + (Deviation)/(2*sqrt(Nearest-Perfect-Square))`

The nearest perfect square to 229 is `225`. And the deviation is `229 - 225 = 4`

`sqrt 229 approx sqrt 225 + (4)/(2*sqrt225)`

`approx 15 + (4/30) approx color(blue)(15.133`

Hence we get `sqrt458 approx 1.4142 * 15.133 approx color(green)(21.4015`