Square Roots

What is a Square Root?

The square root is the inverse operation of squaring. If $n^2 = x$, then the square root of $x$ is $n$. We denote it by the symbol $\sqrt{x}$.

Finding Square Roots

Method 1: Repeated Subtraction

Since a square is the sum of consecutive odd numbers, we can subtract odd numbers ($1, 3, 5\dots$) from the given number until we reach zero. The number of steps is the square root.

Example: Find $\sqrt{25}$

1. $25 - 1 = 24$ (Step 1)
2. $24 - 3 = 21$ (Step 2)
3. $21 - 5 = 16$ (Step 3)
4. $16 - 7 = 9$ (Step 4)
5. $9 - 9 = 0$ (Step 5)

We reached 0 in 5 steps. Therefore, $\sqrt{25} = 5$.

Method 2: Prime Factorization

For larger numbers, we use prime factorization. We pair identical prime factors and pick one from each pair.

Example: Find $\sqrt{324}$

1. Find prime factors of 324: $324 = 2 \times 2 \times 3 \times 3 \times 3 \times 3$
2. Group them in pairs: $324 = (2 \times 2) \times (3 \times 3) \times (3 \times 3)$
3. Take one from each pair: $\sqrt{324} = 2 \times 3 \times 3$
4. Multiply: $2 \times 3 \times 3 = 18$

Method 3: Estimation

If a number is not a perfect square (or is very large), we can estimate. Example: Find $\sqrt{250}$

1. We know $10^2 = 100$ and $20^2 = 400$. So the root is between 10 and 20.
2. $15^2 = 225$ and $16^2 = 256$.
3. 250 is closer to 256 than 225.
4. Therefore, $\sqrt{250} \approx 16$.