Root Calculator
Understanding n-th Roots — From Definition to Hand-Calculation Tips
Quick recap:
The n-th root of a positive number a is the single positive value b that, when raised to the power n, recreates a.
\( b^n = a\quad\Longleftrightarrow\quad b=\sqrt[n]{,a,}\)
Below you’ll find a refreshed, reader-friendly guide that
defines square, cube, and general n-th roots,
walks through a “no-calculus” pencil-and-paper procedure for finding them, and
gives fully-worked numeric examples you can copy in class or the kitchen.
1 What do we mean by “root”?
| Name | Symbol | Verbal read-out | Equation |
|---|---|---|---|
| Square root | \( \sqrt{a}, \) | “root of a” (power 2) | \( b^2=a \) |
| Cube root | \( \sqrt[3]{a} \) | “cube root” (power 3) | \( b^3=a \) |
| n-th root | \( \sqrt[n]{a} \) | “n-th root” | \( b^n=a \) |
Roots answer the inverse question of exponentiation: “What base, repeated n times, rebuilds my number?”
2 Why roots matter
Geometry – side length of a square with area A is \( \sqrt{A}\).
Finance – annual compound growth rate for n years is \( \sqrt[n]{\text{total multiplier}} – 1\).
Chemistry & Physics – reaction half-life, RMS voltage, sound intensity averages all hide a root.
3 A pencil-and-paper algorithm (no calculus required)
The time-honoured approach is a special case of Newton’s method. For square roots it often goes by the Babylonian method; for higher powers we tweak the averaging step.
3.1 Square root steps
Guess a starting value \(b\).
Divide: \(c = a / b\).
Average: \(b_{\text{new}} = (b + c)/2\).
Loop to step 2 until the digits stop changing at your desired precision.
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Example – √27 (to 3 decimals)
| Step | b (guess) | c = 27 / b | New average |
|---|---|---|---|
| 1 | 5.125 | 5.268 | 5.1965 |
| 2 | 5.1965 | 5.1960 | 5.1962 |
| 3 | 5.1962 | 5.1962 | 5.196 |
Final answer ≈ 5.196.
3.2 General n-th root steps
Guess \(b\).
Divide: \(c = a / b^{,n-1}\).
Average \[ b_{\text{new}} ;=; \frac{(n-1),b + c}{n} \]
Repeat step 2 until stable.
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Example – 8-th root of 15 (to 3 decimals)
| Step | b | c = 15 / b⁷ | New average |
|---|---|---|---|
| 1 | 1.432 | 1.405 | 1.388 |
| 2 | 1.388 | 1.403 | 1.402 |
| 3 | 1.402 | 1.403 | 1.403 |
Rounded to three decimals, \( \sqrt[8]{15} \approx 1.403 \).
Why it works: each cycle corrects the guess by the slope of \(x^n-a\); mathematically this is Newton-Raphson, but you only need division and averaging on paper.
4 Handy reference table (perfect roots)
| a | √a | ³√a | ⁴√a |
|---|---|---|---|
| 4 | 2 | 1.587 | 1.414 |
| 8 | 2.828 | 2 | 1.682 |
| 16 | 4 | 2.520 | 2 |
| 27 | 5.196 | 3 | 2.279 |
| 32 | 5.657 | 3.174 | 2.378 |
| 64 | 8 | 4 | 2.828 |
| 81 | 9 | 4.327 | 3 |
(Values rounded to 3 dp where necessary.)
5 Calculator-free tips & tricks
| Situation | Shortcut |
|---|---|
| Area → side | Estimate by bracketing the perfect squares surrounding the area. 81 cm² means the side is just under 9 cm. |
| Large n | Convert to scientific notation: ⁸√(3.2 × 10⁷) = ⁸√3.2 × 10^(7/8). |
| Back-of-envelope growth | Doubling time ≈ ln 2 / ln (1+rate) uses natural logs (another inverse function!) instead of roots. |
Key takeaways
Roots undo powers.
The Babylonian/Newton averaging trick converges fast—even on a scratch pad.
The same template works for any exponent after you adjust the averaging formula.
Mastering a few perfect powers lets you ball-park answers in your head before you ever hit the √ button.