How do you simplify the square root of 458?

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Answer 1 · 2015-04-23T08:27:45

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 $sqrt458 approx 1.4142 * 15.133 approx color(green)(21.4015$