Square Root Calculator

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This calculator can be used to find the approximate value of the square root of a positive number. If the number is an integer smaller than 10 trillion, it will also provide the simplified form of the radical. To use the calculator, provide a value and click the "Calculate" button.


Root and square root

The nth root of a number is denoted using the following notation:

nx = r

where n is the index, x is the radicand, and r is the nth root. The nth root of a number is the number r, that when raised to the index n, is equal to x. In other words, we can rewrite the above expression as:

rn = x

The square root is a specific case in which n = 2, and is the most commonly used root, though n can be any integer. It is important to note that in cases where n is not specified, the root is assumed to be the square root by convention. For any other root, such as a cubed root (n = 3), n will be specified.

As an example, √4, read as the "square root of four," is equal to ±2, since (±2)2 = 4. Thus, the process of finding the square root of a number involves determining what number, when multiplied by itself (squared), yields the value under the radical symbol. Note that the square root of every positive real number has two solutions, a negative and a positive one. This is because when any real number is squared, it is positive, since a positive number multiplied by a positive number is positive and a negative number multiplied by a negative number is also positive.

Simplify square roots

Simplifying square roots involves trying to decompose the radicand into a product that includes perfect squares; if this cannot be done, the square root is in simplified form. Perfect squares are numbers that have integer square roots. For example, 4 is a perfect square because its square root is 2; 16 is a perfect square because its square root is 4; 144 is a perfect square because its square root is 12, and so on. If the radicand can be rewritten as a product that includes perfect squares, the perfect squares can be pulled out from under the radical symbol using the following property.

a × b = √a × √b

One way to determine whether a number can be rewritten as a product that includes a perfect square is to determine the prime factorization of the number and identify pairs of prime factors. For example:

76 = √2 × 2 × 19 = √22 × 19 = 2√19

In the above example, the pair of 2s forms the perfect square 4, so it can be simplified to 2, and since 19 is a prime number, the radical 2√19 cannot be simplified any further.

Square root addition, subtraction, multiplication, and division

Adding and subtracting square roots requires that the radicand be exactly the same. This is similar to the concept of common denominators when adding and subtracting fractions. For example,

5 + √7 = √5 + √7

It is not possible to simplify this any further because the radicands are not the same. Simplifying this expression further typically requires the use of a calculator or computer. However, if the radicands are the same, simply add/subtract the numbers outside the radical symbol:

7√5 + 12√5 - 2√5 = (7 + 12 - 2)√5 = 17√5

Multiplying square roots involves simply multiplying the values under the radical symbol. This is because of the following property:

a × √b = √a × b

For example:

5 × √7 = √5 × 7 = √35

Dividing square roots makes use of the following property:

Thus, simply divide the radicands and simplify if possible. For example:

Like the case of multiplication, if there are any numbers outside of the radicand, just divide them separately.

Estimating a square root

Estimating square roots is a tedious process that should ideally be done using a calculator. In cases where this is not possible, use the following algorithm:

  1. Identify the perfect squares between which the number lies.
  2. Divide the number by the square root of one of the perfect squares surrounding it.
  3. Find the average of the result and the square root used to divide the number; the result is the first estimate of the square root.
  4. Use the average to divide the number, then find the new average between this number and the previous average. Repeat this process, dividing the number by the new average each time, then finding the average of the result and the previous average. Each repetition will result in a more accurate estimate of the square root.

For example, estimate √8:

  1. 8 lies between the perfect squares 4 and 9.
  2. The square root of 4 is 2, and the square root of 9 is 3. Selecting 3, 8/3=2.667.
  3. (2.667+3)/2=2.834; 2.8342 = 8.032
  4. 8/2.834=2.823; (2.834+2.823)/2=2.829; 2.8292=8.003

The more this process is repeated, the more accurate the estimate becomes, but in this case, the estimate is off by only 0.003 after only one repetition. Just for reference, using a calculator, √8=2.82842712475.

Square root of negative numbers

As mentioned above, the square root of a positive real number has two solutions: a negative and a positive one. This fact implies that the square root of a negative number cannot have a real root, since it is not possible for the square of a real number to be negative. However, there are cases in which it is necessary to compute the square root of a negative number, so imaginary numbers were created. The imaginary number i is defined as follows:

i2 = -1

i = √-1

Since any negative number can be written as a product of that number and -1, the following property,

a × √b = √a × b,

can be used to find the square root of a negative number as follows:

-16 = √16 × (-1) = √16 × √-1 = 4i

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