Roots Are Fractional Exponents
The nth root of x is the number that, raised to the power n, gives x. It is the inverse of exponentiation, and it is the same operation as a fractional exponent:
n√x = x1/n
This equivalence is what makes roots calculable at all — every root is computed as a power with a fractional exponent.
Roots and Powers Chart
| n | √n | ³√n | n² | n³ |
|---|---|---|---|---|
| 2 | 1.4142 | 1.2599 | 4 | 8 |
| 3 | 1.7321 | 1.4422 | 9 | 27 |
| 4 | 2.0000 | 1.5874 | 16 | 64 |
| 5 | 2.2361 | 1.7100 | 25 | 125 |
| 8 | 2.8284 | 2.0000 | 64 | 512 |
| 10 | 3.1623 | 2.1544 | 100 | 1,000 |
| 16 | 4.0000 | 2.5198 | 256 | 4,096 |
| 25 | 5.0000 | 2.9240 | 625 | 15,625 |
| 100 | 10.0000 | 4.6416 | 10,000 | 1,000,000 |
Simplifying Radicals
A radical simplifies when the number under it contains a perfect square factor. √72 becomes 6√2, because 72 = 36 × 2 and √36 = 6. The process: find the largest perfect square that divides the number, take its root outside, leave the rest inside.
Simplified form is exact where a decimal is not — 6√2 is precisely correct, while 8.485 is rounded. This matters in algebra and geometry, where exactness carries through subsequent steps.
Even and Odd Roots
Odd roots work for negative numbers: the cube root of −8 is −2, because (−2)³ = −8. Even roots of negative numbers have no real value, since any real number squared is positive. They exist in the complex numbers, where √−1 is defined as i.
A second subtlety: x² = 9 has two solutions, +3 and −3, but the radical symbol √9 denotes only the positive one. This is why quadratic solutions are written with ± explicitly.
Estimating Roots Mentally
Find the nearest perfect squares and interpolate. For √72: 8² = 64 and 9² = 81, so the answer lies between 8 and 9, and closer to 8.5 since 72 sits near the middle. The true value is 8.485 — close enough for most sanity checks.
Frequently Asked Questions
Why can't I take the square root of a negative number?
Because no real number multiplied by itself gives a negative result. Complex numbers extend the system specifically to allow it.
What is the difference between √x and x0.5?
Nothing — they are the same operation written two ways. The fractional exponent form generalises more easily to other roots.
Is √2 rational?
No, and the proof of that is one of the oldest results in mathematics. Its decimal expansion never terminates or repeats.