Choose square root, cube root or any other root, type the number and get the answer to ten decimal places. Where the root does not come out exactly, the calculator gives the simplest surd form (such as √72 = 6√2) and shows how to estimate it by hand in a few steps.
Choose square root, cube root or any root.
For any root, enter n (4 for a fourth root, and so on).
Type the number.
Read the root, its surd form and the steps.
Definition: ⁿ√a = b means bⁿ = a
As a power: ⁿ√a = a^(1 ÷ n)
Simplifying: √(p² × q) = p√q
Estimate (square root): new guess = (guess + a ÷ guess) ÷ 2
A root is the opposite of raising a number to a power. Squaring 9 gives 81, so the square root of 81 is 9. Cubing 4 gives 64, so the cube root of 64 is 4. The root calculator finds the square root, the cube root or any higher root of the number you enter, to ten decimal places.
Two extra results make it useful for study. When the root is not a whole number, it gives the simplest surd form, the way answers are expected in Class 9 and 10 algebra, for example root 72 written as 6 root 2. It also shows how to estimate the root on paper by guessing, dividing and averaging, a method that reaches three correct decimals in a few rounds. This helps in exams where calculators are not allowed and gives a feel for the size of the answer.
1. Check whether the number is a perfect square or cube, such as 144 or 125. If so, the root is a whole number.
2. If not, break the number into prime factors.
3. For a square root, take out one factor for every pair; for a cube root, one for every group of three.
4. Multiply what comes out. What is left stays under the root sign.
5. For a decimal value, guess a number, divide the original by the guess, and average the two.
6. Repeat the last step with the new value until the digits stop changing.
Taking the nth root is the same as raising to the power one over n. The square root of a is a to the power one half, and the cube root is a to the power one third. This is why the rules of powers also work for roots: the root of a product is the product of the roots, so root 72 equals root 36 times root 2. That single rule is the basis of simplifying surds.
A square, a fourth power or any even power of a real number is never negative, so a negative number has no real square root or fourth root. Odd powers keep the sign, so odd roots of negative numbers are fine: the cube root of minus 8 is minus 2. A positive number has two square roots, one positive and one negative, and the root sign by itself always means the positive one.
If your guess for a square root is too small, dividing the number by it gives a result that is too big, and the true root lies between the two. Their average is therefore a better guess. Repeating this doubles the number of correct digits each time. The method is thousands of years old, was used in ancient Babylon and India, and is still what calculators and computers use inside, under the name Newton's method.
Kiran needs the side of a square plot whose area is 200 square metres, both as a simplified surd for his maths homework and as a decimal for fencing.
What the square root means: Find b so that b multiplied by itself 2 times gives 200.
Take out perfect powers: 200 = 10² × 2 √200 = 10√2
Estimate by hand: guess, divide, average: Guess 14: 200 ÷ 14 = 14.285714 New guess = (14 + 14.285714) ÷ 2 = 14.142857
Improve the guess (round 2): Guess 14.142857: 200 ÷ 14.142857 = 14.141414 New guess = (14.142857 + 14.141414) ÷ 2 = 14.142136
Improve the guess (round 3): Guess 14.142136: 200 ÷ 14.142136 = 14.142136 New guess = (14.142136 + 14.142136) ÷ 2 = 14.142136
Check: 14.14213562^2 = 200
Answer: Square root 14.1421356237; Simplest surd form 10√2; Both square roots ±14.142136
Halving the number instead of taking its square root: the square root of 16 is 4, not 8.
Writing the root of a sum as the sum of the roots. Root of 9 + 16 is 5, not 3 + 4.
Leaving a surd unsimplified, such as root 50 instead of 5 root 2.
Saying a negative number has no cube root. It does: the cube root of −27 is −3.
Forgetting the negative square root when solving x² = 49, which has two answers, 7 and −7.
Finding the side of a square from its area or the edge of a cube from its volume.
Simplifying surds in algebra and trigonometry.
Working out the diagonal of a rectangle or screen with Pythagoras theorem.
Calculating yearly growth rates from a total growth over several years.
Standard deviation and other statistics that need a square root.
Can I take the square root of a negative number?
Not among real numbers, because any real number squared is positive or zero. Cube roots and other odd roots of negative numbers do exist: ∛(−27) = −3.
How do I simplify √72?
Find the largest perfect square that divides 72, which is 36. Then √72 = √36 × √2 = 6√2.
Why does a positive number have two square roots?
Both 7 × 7 and (−7) × (−7) equal 49. The symbol √ means the positive one, called the principal root.
How do I find a square root by hand?
Guess a value, divide the number by your guess, and take the average of the guess and the result. Repeat with the new value. Two or three rounds usually give three correct decimals.