# What is the square root of 729?

May 23, 2018

The positive square root is $27$ and the negative one $- 27$.

#### Explanation:

First find the prime factorisation of $729$:

$\textcolor{w h i t e}{000} 729$
$\textcolor{w h i t e}{000} \text{/"color(white)(0)"\}$
$\textcolor{w h i t e}{00} 3 \textcolor{w h i t e}{00} 243$
$\textcolor{w h i t e}{00000} \text{/"color(white)(0)"\}$
$\textcolor{w h i t e}{0000} 3 \textcolor{w h i t e}{000} 81$
$\textcolor{w h i t e}{0000000} \text{/"color(white)(00)"\}$
$\textcolor{w h i t e}{000000} 3 \textcolor{w h i t e}{000} 27$
$\textcolor{w h i t e}{000000000} \text{/"color(white)(00)"\}$
$\textcolor{w h i t e}{00000000} 3 \textcolor{w h i t e}{0000} 9$
$\textcolor{w h i t e}{000000000000} \text{/"color(white)(0)"\}$
$\textcolor{w h i t e}{00000000000} 3 \textcolor{w h i t e}{000} 3$

So:

$729 = {3}^{6}$

Hence:

$\sqrt{729} = \sqrt{{3}^{6}} = {3}^{3} = 27$

This is the positive square root.

$729$ also has a negative square root $- 27$