Finding $\sqrt{625}$ without a calculator

Question

We were factoring $25n^2+15n-4$ in class and my professor used the quadratic formula. This root came up and he wrote the answer directly. He doesn't want us to use calculators.

Is there any trick for this?

Answer 1

Hint: Remark that $625=5^4$, and take the square root..

Answer 2

$20^2=400<625<900=30^2$, so $625=(2 \_)^2$. The only integer $x$ between $0$ and $9$ such that $x^2$ ends with $5$ is $5$, so if we try $25$, we come with $(25)^2=625$.

Answer 3

I guess your professor has been using the following trick: the square of a number of the form $10n + 5$ is $100n(n+1) + 25$. Indeed $(10n + 5)^2 = 100n^2 + 100n + 25 = 100n(n+1) + 25$. For instance: \begin{align} n &= 0: && 5^2 = 25 &&\text{since $0 \times 1 = 0$} \\ n &= 1: && 15^2 = 225 &&\text{since $1 \times 2 = 2$} \\ n &= 2: && 25^2 = 625 &&\text{since $2 \times 3 = 6$} \\ n &= 3: && 35^2 = 1225 &&\text{since $3 \times 4 = 12$} \\ n &= 4: && 45^2 = 2025 &&\text{since $4 \times 5 = 20$} \\ n &= 5: && 55^2 = 3025 &&\text{since $5 \times 6 = 30$} \\ &\ \vdots \end{align} If you know this trick, it is not difficult to remember that $25^2 = 625$...

Answer 4

HINT: Decompose 625 into prime factors to see how you can find $\sqrt{625}$ without a calculator.

Prime factorization helps makes calculations nicer if you extract the square from the square root. For example, $\sqrt{8} = \sqrt{2^3} = 2\sqrt{2}$. Most of the people find $2\sqrt{2}$ much nicer to work with than $\sqrt{8}$.