Wren writes √(x + 3) = x - 3 on the chalkboard wall. "Solve it. Then tell me what you notice."
Comet squares both sides. "x + 3 = x² - 6x + 9. So x² - 7x + 6 = 0. That factors: x = 1 or x = 6."
"Check them," Wren says.
"x = 6: root of 9 is 3, and 6 minus 3 is 3. Good. x = 1: root of 4 is 2, but 1 minus 3 is -2."
Nova pulses softly over the 1. "Would you like a hint? A square root is never negative."
"So 2 cannot equal -2. x = 1 is extraneous," Comet says. "Squaring made it up, like clearing a denominator did last week."
"Now write √x as a power," Wren says. "x to the one half. Roots are fraction exponents in disguise."
For √(x + 3) = x - 3, squaring gives x + 3 = (x - 3)². Collect terms: x² - 7x + 6 = 0.
The solutions of that quadratic are x = 1 and x = 6. Check x = 1: √4 = 2, but 1 - 3 = -2.
A square root is never negative, so x = 1 is extraneous. Check x = 6: √9 = 3 and 6 - 3 = 3. It works.
An equation like √(x + 3) = -2 has no solution at all. Squaring gives x = 1, but the check fails, because a root cannot be negative.
Sometimes the check comes first. In √(x + 3) = x - 3, the left side is never negative. So x - 3 must be 0 or more.
That means x must be at least 3 before you even start. The value 1 was never a candidate.
Both ways reach the same answer. Way one is mechanical and safe. Way two is quicker when you see it.
A root is a fraction power. √x is x^(1/2), ∛x is x^(1/3), and ∛(x²) is x^(2/3). The bottom is the root, the top is the power.
So 8^(2/3) means the cube root of 8, then squared: 2² = 4. And 16^(3/4) = (⁴√16)³ = 2³ = 8.
The exponent rules still hold. x^(1/2) × x^(1/2) = x^(1/2 + 1/2) = x. That is exactly why √x × √x = x.
Simplify roots by pulling out perfect squares: √48 = √16 × √3 = 4√3, and √72 = 6√2.
| Equation | Way one: square and check | Way two: think first | Solution |
|---|---|---|---|
| √(x + 3) = x - 3 | x = 1 or 6, then the check rejects 1 | x - 3 must be 0 or more, so x is at least 3 | x = 6 |
| √(x + 3) = -2 | x = 1, then the check fails | a root is never negative | no solution |
| √(4t + 1) = 5 | t = 6, and the check passes | the right side is positive, so squaring is safe | t = 6 |
| Statement | True or false? |
|---|---|
| A square root of a number is never negative. | ? |
| 8^(2/3) = 4 | ? |
| 16^(3/4) = 12 | ? |
| √(x + 3) = -2 has the solution x = 1. | ? |
| 4 × 4 × 3 = 48 | ? |
Excellent. Tomorrow the stopwatch comes out in the Glass House Lab and your own drip table gets an inverse.