Square Roots by Factorisation
Break 144 into prime factors. Pair them up to find the root.
Equation
y = sqrt(1 * x)
Graph
Table
| x | y |
|---|---|
| 0 | 0 |
| 2 | 1.41 |
| 4 | 2 |
| 6 | 2.45 |
| 8 | 2.83 |
| 10 | 3.16 |
| 12 | 3.46 |
| 14 | 3.74 |
| 16 | 4 |
What this lesson covers
What you do
You shape the function y = sqrt(k * x) and watch the graph answer.
Challenges to clear
- √144 = √(9 · 16): slide k to 9 and check the x = 16 row shows 12 — pull the pairs out: 3 · 4.
- Same pairing, smaller: √(9 · 4) = 3 · 2 = 6 at x = 4.
- √400 splits into a pair of factors. Slide m to 4 — one of them.
- Now find the other: at x = 4 the row must read 20. Pull one number out of each pair — 5 × 2 × 2 = 20.
Check yourself
Why do we pair prime factors when finding a square root?
If a prime factor appears only once (e.g., 50 = 2 × 5 × 5), what happens to that lone factor?
- Because a square root 'unzips' pairs of identical factors, allowing one from each pair to exit the radical. — correct
- Because adding the factors together gives the correct sum.
- Because square roots always result in a number smaller than half the original.
- Because unpaired factors disappear completely.
- It remains inside the square root symbol. — correct
- It comes out as a decimal (e.g., 2.5).
- It gets squared automatically.
- It disappears because it has no pair.