# What is the square root of 2025?

May 31, 2015

We could solve this by factoring:

$2025$
$\textcolor{w h i t e}{\text{XXXXX}}$$= 5 \times 405$
$\textcolor{w h i t e}{\text{XXXXX}}$$= 5 \times 5 \times 81$
(maybe at this point we recognize $81 = {9}^{2}$, but let's continue pretending we don't)
$\textcolor{w h i t e}{\text{XXXXX}}$$= 5 \times 5 \times 3 \times 27$
$\textcolor{w h i t e}{\text{XXXXX}}$$= 5 \times 5 \times 3 \times 3 \times 9$
$\textcolor{w h i t e}{\text{XXXXX}}$$= 5 \times 5 \times 3 \times 3 \times 3 \times 3$
and we have completely factored the given value.

Group the factoring in pairs of equal value:
$\textcolor{w h i t e}{\text{XXXXX}}$$= \textcolor{red}{5 \times 5} \times \textcolor{g r e e n}{3 \times 3} \times \textcolor{b l u e}{3 \times 3}$
$\textcolor{w h i t e}{\text{XXXXX}}$$= \textcolor{red}{{5}^{2}} \times \textcolor{g r e e n}{{3}^{2}} \times \textcolor{b l u e}{{3}^{2}}$
$\textcolor{w h i t e}{\text{XXXXX}}$$= {\left(\textcolor{red}{5} \cdot \textcolor{g r e e n}{3} \cdot \textcolor{b l u e}{3}\right)}^{2}$
$\textcolor{w h i t e}{\text{XXXXX}}$$= {45}^{2}$

If $2025 = {45}^{2}$
then
$\textcolor{w h i t e}{\text{XXXXX}}$$\sqrt{2025} = 45$