# How do you find the square root of 54?

Sep 12, 2015

$\sqrt{54} = 3 \sqrt{6} \approx 7.34846922835$

#### Explanation:

If $a , b \ge 0$ then $\sqrt{a b} = \sqrt{a} \sqrt{b}$, so

$\sqrt{54} = \sqrt{{3}^{2} \cdot 6} = \sqrt{{3}^{2}} \sqrt{6} = 3 \sqrt{6}$

If you want an approximate value, then so long as you have memorised approximate values for $\sqrt{2}$ and $\sqrt{3}$ then it's just $3 \sqrt{2} \sqrt{3}$

From memory:

$\sqrt{2} \approx 1.414213562373$
$\sqrt{3} \approx 1.73205080756$

Alternatively you could use a Newton Raphson type method, such as the one I describe in http://socratic.org/questions/how-do-you-find-the-square-root-28

If you started with $7$, this would give you successive rational approximations for $\sqrt{54}$:

$\frac{7}{1} = 7.0$

$\frac{103}{14} \approx 7.36$

$\frac{21193}{2884} \approx 7.34847$

$\frac{898285873}{122241224} \approx 7.34846922835$