Root Calculator
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General Root Calculator
Roots in Mathematics
In mathematics, the general root, or the nth root of a number a, is another number b that, when multiplied by itself n times, equals a. In equation format:
n√a = b and bn = a
Estimating a Root
Some common roots include the square root, where n = 2, and the cube root, where n = 3. Calculating square roots and nth roots by hand is fairly intensive — it requires estimation and trial and error. More precise and efficient methods exist, but the one below needs no complicated mathematics. To calculate √a:
- Estimate a number b.
- Divide a by b. If the number c returned is precise to the desired decimal place, stop.
- Average b and c and use the result as a new guess.
- Repeat step two.
Guess: 5.125
27 ÷ 5.125 = 5.268
(5.125 + 5.268) / 2 = 5.197
27 ÷ 5.197 = 5.195
(5.195 + 5.197) / 2 = 5.196
27 ÷ 5.196 = 5.196 → √27 ≈ 5.196
Estimating an nth Root
Calculating nth roots can be done with a similar method, with modifications to deal with n. While computing square roots entirely by hand is tedious, estimating higher nth roots — even using a calculator for the intermediate steps — is significantly more tedious. To calculate n√a:
- Estimate a number b.
- Divide a by bn−1. If the number c returned is precise to the desired decimal place, stop.
- Average: [b × (n−1) + c] / n.
- Repeat step two.
Guess: 1.4
15 ÷ 1.47 = 15 ÷ 10.541 = 1.423
(1.4 × 7 + 1.423) / 8 = 1.4029
15 ÷ 1.40297 = 1.4023
(1.4029 × 7 + 1.4023) / 8 = 1.4028
→ 8√15 ≈ 1.403
It should then be clear that computing any further will keep producing a number that rounds to 1.403, making 1.403 the final estimate to 3 decimal places.
Frequently Asked Questions
What is the nth root of a number?
The nth root of a number a is another number b that, when multiplied by itself n times, equals a: if bn = a, then b is the nth root of a. The square root is the case n = 2 and the cube root is n = 3.
Can you take the root of a negative number?
For odd roots, yes: the cube root of −27 is −3, because (−3)3 = −27. For even roots, no real answer exists — no real number squared or raised to the 4th power gives a negative — so the square root of a negative number is not a real number.
How do you estimate a square root by hand?
Use the averaging method: guess a number b, divide the radicand a by b to get c, then average b and c and use the result as the new guess. Repeat until the guess stops changing at the precision you need. Estimating √27 from a guess of 5.125 settles at 5.196 in a few rounds.
How are roots and exponents related?
A root is a fractional exponent: the nth root of a is a1/n. That is why this calculator can verify every answer by raising it back to the power n, and why a negative degree such as −2 gives the reciprocal: the degree −2 root of 16 is 1/√16 = 0.25.