Find the square root of any number — the value that, when multiplied by itself, gives your number.
The square root is the number that produces your input when multiplied by itself.
If the result is a whole number, the input is a perfect square.
Negative numbers have no real square root, because any real number multiplied by itself is positive.
Square root: √n = x where x × x = n
Also written: n^(1/2)
Finding the square root of 144.
Number: 144
Look for a number that multiplied by itself gives 144
12 × 12 = 144
144 is so a perfect square
Answer: √144 = 12
A square root calculator finds the non-negative number that, multiplied by itself, gives your input. It also tells you whether the input is a perfect square. The most common everyday question it answers is 'if a square has this area, how long is each side?'
In India, plot and flat sizes are quoted in square feet or square yards, so a buyer who hears '1,200 sq ft' may want to picture it as a square of about 34.6 ft on each side. Students need square roots for Pythagoras' theorem in Class 7 to 10, for the quadratic formula, for standard deviation in statistics and for physics formulas such as v = √(2gh). The calculator gives the principal, positive root.
1. Check the number is not negative; negative numbers have no real square root.
2. Try prime factorisation: pair up equal prime factors and take one from each pair. For example, 576 = 2⁶ × 3², so √576 = 2³ × 3 = 24.
3. If the number is not a perfect square, find the two perfect squares it lies between to get a first estimate.
4. Refine the estimate x with the averaging rule: new x = (x + n ÷ x) ÷ 2.
5. Repeat until two successive estimates agree to the precision you need.
6. Check by squaring the answer.
Both 12 and −12 square to 144, so the equation x² = 144 has two solutions. The symbol √ is defined to mean only the non-negative one, called the principal square root. That is what this calculator shows. When you solve an equation such as x² = 144, you must write x = ±√144 = ±12 yourself. When the square root represents a length, such as the side of a plot, only the positive value makes sense, which is why the convention is useful.
If x is a guess for √n, then n ÷ x is on the other side of the true root: when x is too big, n ÷ x is too small, and vice versa. Their average lies closer to the root than either. This is the Babylonian method, a special case of Newton's method, and each round roughly doubles the number of correct digits. Starting from 35 for √1200 gives (35 + 34.286) ÷ 2 = 34.643, already within 0.002 of the true value 34.641.
If n is a whole number but not a perfect square, √n is irrational. Its decimal never ends or repeats, so any displayed value is rounded. In school algebra you are often asked for an exact simplified surd instead. Take out the largest square factor: √1200 = √(400 × 3) = 20√3. The decimal 34.641... and the surd 20√3 are the same number, but only the surd is exact. The calculator gives the decimal; the surd form is for your written answer.
Deepak is looking at a 1,200 sq ft residential plot in Hyderabad and wants to picture how long each side would be if the plot were a perfect square.
Answer: Square root 34.6410161514; Perfect square? No; Check 34.641016 × 34.641016 = 1,200
Forgetting the ± when solving x² = k in algebra.
Taking the square root of each term separately, as in √(9 + 16) = 3 + 4; the correct value is √25 = 5.
Assuming a plot of a given area must be square, when real plots are usually rectangles with different sides.
Mixing units, such as taking the root of square yards and reading the answer in feet.
Estimating the side of a square plot, room or tile from its area.
Finding the hypotenuse with Pythagoras' theorem in geometry and construction.
Solving quadratic equations with the quadratic formula.
Computing standard deviation from variance in statistics.
Physics formulas such as speed after a free fall, v = √(2gh).
Why can't I take the square root of a negative number?
Because a positive number times a positive number is positive, and a negative number times a negative number is also positive. No real number, when squared, gives a negative answer. Mathematicians use imaginary numbers to handle this, but there is no real answer.
Doesn't every number have two square roots?
Yes — both 12 and −12, when squared, give 144. By agreement, the √ symbol means the positive root, which is what this calculator shows.
What is a perfect square?
A number whose square root is a whole number — like 1, 4, 9, 16, 25, 36, and so on.