The Hidden Square Root
Simplify surds by hunting for perfect squares. Turn messy roots into clean multiples.
Equation
y = 1^2 * x
Graph
Table
| x | y |
|---|---|
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
| 4 | 4 |
| 5 | 5 |
| 6 | 6 |
| 7 | 7 |
| 8 | 8 |
| 9 | 9 |
| 10 | 10 |
What this lesson covers
What you do
You shape the function y = a^2 * x and watch the graph answer.
Challenges to clear
- (a√5)² = a²·5: slide a until the x = 5 row shows 45 — then √45 = a√5 with that a.
- Same a, another surd: the x = 2 row shows 18, so √18 = 3√2 as well!
- Simplify √150 into a√b. Slide a to the largest whole number whose square divides 150.
- Now find what is left under the root: slide b until the x = 1 row reads 150 — so √150 = 5√6.
Check yourself
Why is √45 equal to 3√5?
- Because 45 is divisible by 3.
- Because 45 = 9 × 5, and √9 = 3, so √45 = √9 × √5 = 3√5. — correct
- Because 3 × 5 = 15, and 15 is close to 45.
- Because √45 is approximately 6.7, and 3√5 is approximately 6.7, so they are just estimates.
Think about it
- √45 = a√5 means (a√5)² = a²·5 must equal 45. What a makes a²·5 = 45?