Unsquaring 144: Prime Factorisation
Pair up prime factors to find the square root without a calculator.
Equation
y = sqrt(1^2 * 2^2 * x)
Graph
Table
| x | y |
|---|---|
| 0 | 0 |
| 1 | 2 |
| 2 | 2.83 |
| 3 | 3.46 |
| 4 | 4 |
What this lesson covers
What you do
You shape the function y = sqrt(a^2 * b^2 * x) and watch the graph answer.
Challenges to clear
- Factor pairs: set a = 2 and b = 6 — √(2² · 6²) = 2 · 6 = 12 at x = 1.
- x = 4 doubles it to 24. Any pair with a·b = 12 gives the same root — that is why factorisation works.
- √225 this time. Take out the first prime pair: slide a to 3.
- Now the rest: x = 1 must read 15, so b is 5. Every pair of equal primes sends exactly one of itself outside the root.
Check yourself
Why does grouping prime factors in pairs allow us to find the square root easily?
If the prime factorisation of a number has one '3' left over (unpaired), what does that mean about the square root?
- Because sqrt(a*a) equals a, so each pair simplifies to a single number. — correct
- Because multiplying primes is faster than adding them.
- Because odd numbers cannot be square roots.
- Because 144 is divisible by 2 and 3.
- The number is not a perfect square, so its root will be irrational. — correct
- You made a mistake in your division.
- The square root is exactly 3.
- You should multiply the 3 by 2 to make it even.