What is the square root of 407

The short answer is \( \sqrt{ 407 } = 20.174241001832 \).

If that's all you're looking for, thanks for coming by. But if you're interested in the hows and whys behind square roots, continue reading and we'll tell you all about the square root of 407.

407 is not a perfect square

Let's look at some math

$$ \LARGE \sqrt{ 407 } = 20.174241001832 $$

Note that \(20.174241001832\) is not a whole number, therefore 407 is not a perfect square.

The next perfect square greater than 407 is 441. The previous perfect square less than 407 is 400.

Square root of 407 as an exponent

Any square root can be converted to a number with a fractional exponent. In the case of 407 the following two values are equal.

$$ \LARGE \sqrt{ 407 } = 407^{^1/_2} $$

Square root of 407 as a fraction

Since 407 is not a perfect square, sometimes we might want to get a fractional estimate of the square root value. In the case of 407 that value is \(\Large 20 \frac{23}{132}\).

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