Root Calculator
Calculate square roots, cube roots, and general nth roots of numbers. Get precise results for your mathematical calculations.
Square Root Calculator
Cube Root Calculator
General Root Calculator
Understanding 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:
bn = a
Common Roots
| Root Name | Notation | Example | Result |
|---|---|---|---|
| Square Root | √ or ²√ | √25 | 5 |
| Cube Root | ∛ or ³√ | ∛27 | 3 |
| Fourth Root | ∜ or ⁴√ | ⁴√16 | 2 |
| Fifth Root | ⁵√ | ⁵√32 | 2 |
Estimating a Root
Some common roots include the square root, where n = 2, and the cubed root, where n = 3. Calculating square roots and nth roots is fairly intensive. It requires estimation and trial and error. There exist more precise and efficient ways to calculate square roots, but below is a method that does not require a significant understanding of more complicated math concepts. 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
Example:
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
Final result: 5.196
Estimating an nth Root
Calculating nth roots can be done using a similar method, with modifications to deal with n. While computing square roots entirely by hand is tedious. Estimating higher nth roots, even if using a calculator for intermediary steps, is significantly more tedious. For a simpler, but less efficient method, continue to the following steps and example. 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
Example:
Guess: 1.432
15 ÷ 1.4327 = 1.405
(1.432 × 7 + 1.405)/8 = 1.388
15 ÷ 1.3887 = 1.403
(1.403 × 7 + 1.388)/8 = 1.402
It should then be clear that computing any further will result in a number that would round to 1.403, making 1.403 the final estimate to 3 decimal places.
Related Calculators
Root Calculation Quick Facts
- •The square root of a number is the value that when squared gives the original number
- •Negative numbers don't have real square roots but have complex roots
- •Cube roots can be calculated for negative numbers (∛-8 = -2)
- •Higher roots (4th, 5th, etc.) follow similar patterns to square and cube roots
Frequently Asked Questions
What's the difference between a root and an exponent?
Roots and exponents are inverse operations. While exponents represent repeated multiplication (e.g., 2³ = 2 × 2 × 2 = 8), roots "undo" this operation (e.g., ³√8 = 2). The nth root of a number is the same as raising that number to the power of 1/n.
Can roots be irrational numbers?
Yes, many roots are irrational numbers (numbers that cannot be expressed as a simple fraction). For example, √2 ≈ 1.41421356... is an irrational number that continues infinitely without repeating.
How are roots calculated in computers and calculators?
Modern calculators and computers use sophisticated algorithms to compute roots efficiently. Common methods include Newton's method (an iterative approach) or using logarithms and exponential functions (sincen√a = a1/n = e(ln a)/n).
What are some practical applications of roots?
Roots have many practical applications including in geometry (calculating side lengths), physics (inverse-square laws), finance (compound interest calculations), computer graphics (normalizing vectors), and statistics (standard deviation calculations).
Why can't we take even roots of negative numbers?
In the real number system, even roots (like square roots, fourth roots, etc.) of negative numbers are undefined because no real number multiplied by itself an even number of times will produce a negative result. However, in complex numbers, we can define roots of negative numbers using imaginary numbers (e.g., √-1 = i).