# How do you simplify the square root of 458?

Apr 23, 2015

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.

$\sqrt{458} = \sqrt{2 \cdot 229} = \sqrt{2} \cdot \sqrt{229}$

We know that $\sqrt{2} \approx 1.4142$

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

$\sqrt{N o n - P e r f e c t - S q u a r e} \approx \sqrt{N e a r e s t - P e r f e c t - S q u a r e} + \frac{D e v i a t i o n}{2 \cdot \sqrt{N e a r e s t - P e r f e c t - S q u a r e}}$

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

$\sqrt{229} \approx \sqrt{225} + \frac{4}{2 \cdot \sqrt{225}}$

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

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