Square Roots – Explanation & Examples

Square RootsIn mathematics, a square root of a number x is such that, a number y is the square of x, simplify written as y2 = x.

For example, 5 and – 5 are both square roots of 25 because:

5 x 5 = 25 and -5 x -5 =25.

The square root of a number x is denoted with a radical sign √x or x 1/2. For instance, the square root of 16 is presented as: √16 = 4. A number whose square root is calculated is referred to as radicand. In this expression, √16 = 4 number 16 is the radicand.

What is a Square Root?

Square root is an inverse operation of the squaring a number. In other words, square root is an operation that undoes an exponent of 2.

Properties

  • A perfect square number has a perfect square root.
  • An even perfect number has the square root that is even.
  • Odd perfect number has the square root that is odd.
  • The square root of a negative number is undefined.
  • Only numbers ending with even number of zeros have square roots.

 

How do we find square root of numbers?

There are multiple ways to find the square of the numbers. We will see a few of the here.

Repeated Subtraction

This method involves, successful and repeated subtraction of odd numbers such as 1, 3, 5 and 7 from the number until zero is reached. The square of the number is equal to the number or frequency of subtraction performed on the number

Suppose, we need to calculate the square of a perfect number like 25, the operation is done as:

25 -1= 24
24 -3= 21
21 -5= 16
16 – 7= 9
9 – 9= 0

You can notice that, the frequency of subtraction is 5, therefore the square root of 25 is 5.

Prime Factorization

In this method, a perfect square number is factorized by successive division. The prime factors are grouped into pairs, and the product of each number calculated. The product is therefore, the square root of the number. To find the square of a perfect number such as: 144 is performed as:

  • 144 = 2 × 2 × 2 × 2 × 3 × 3.
  • Pair the prime factors.
  • Selecting one number from each pair.
  • 2 × 2 × 3 = 12.
  • Thus, the √144 = 12.Square Root by Prime Factorization

Division Method

Division method is a suitable technique of calculating the square of a large number. The following are the steps involved:

  • A bar is placed over every pair of digits starting from the right-hand side.
  • Divide left end number by a number whose square is less or equivalent to the numbers under the left end.
  • Take this number as the divisor and quotient. Similarly, take the leftmost number as the dividend
  • Divide to get the result
  • Pull down the next number with a bar to the right-hand side of the remainder
  • Multiply the divisor by 2.
  • To the right of this new divisor, find a suitable dividend. This process is repeated until we get zero as the remainder. The square of the number therefore is equal to the quotient.

The square root of 225 is calculated as

  • Begin the division from the leftmost side.
  • In this case, 1 is our number whose square is below 2.
  • Assigning 1 as the divisor and quotient and multiplying it by 2, gives:
  • Proceed with the steps to get 15 as the quotient.

Practice Questions

1. Which of the following shows the simplified form of $\sqrt{144} + \sqrt{196}$?

2. Which of the following shows the simplified form of $\sqrt{25} \times \sqrt{25}$?

3. Which of the following shows the square root of $1000000$?

4. A school auditorium has $3136$ total number of seats, if the number of seats in the row is equal to the number of seats in the columns. What is the total number of seats in a row.

5. What is the square root of $5625$?

6. A square garden has an area of $16$ squared meters. What is the perimeter of the garden?

7. What is the smallest number that must be added to $570$ to make it a perfect square?

8. What is the square root of the first perfect four-digit number?

9. What is the square root of $0.0025$?


 

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Square Roots – Complete List

Below is the list of Square Roots of various numbers. The solution is just one click away.

Square Root Of 123Square Root Of 89Square Root Of 900Square Root Of 392
Square Root Of 16Square Root Of 1024Square Root Of 8Square Root Of 32
Square Root Of 49Square Root Of 136Square Root Of 576Square Root Of 5
Square Root Of 85Square Root Of 20Square Root Of 500Square Root Of 3125
Square Root Of 121Square Root Of 50Square Root Of 1600Square Root Of 14
Square Root Of 200Square Root Of 96Square Root Of 71Square Root Of 425
Square Root Of 52Square Root Of 53Square Root Of 84Square Root Of 165
Square Root Of 0Square Root Of 676Square Root Of 288Square Root Of 28
Square Root Of 36Square Root Of 112Square Root Of 250Square Root Of 720
Square Root Of 125Square Root Of 441Square Root Of 140Square Root Of 74
Square Root Of 65Square Root Of 116Square Root Of 320Square Root Of 432
Square Root Of 10Square Root Of 73Square Root Of 175Square Root Of 700
Square Root Of 24Square Root Of 1156Square Root Of 9Square Root Of 61
Square Root Of 400Square Root Of 192Square Root Of 98Square Root Of 79
Square Root Of 41Square Root Of 15Square Root Of 6Square Root Of 4000
Square Root Of 225Square Root Of 169Square Root Of 86Square Root Of 91
Square Root Of 12Square Root Of 100Square Root Of 147Square Root Of 663
Square Root Of 1Square Root Of 90Square Root Of 800Square Root Of 216
Square Root Of 18Square Root Of 31Square Root Of 144Square Root Of 220
Square Root Of 256Square Root Of 45Square Root Of 94Square Root Of 243
Square Root Of 23Square Root Of 33Square Root Of 2500Square Root Of 102
Square Root Of 164Square Root Of 841Square Root Of 360Square Root Of 2899
Square Root Of 69Square Root Of 26Square Root Of 168Square Root Of 384
Square Root Of 29Square Root Of 145Square Root Of 148Square Root Of 1133
Square Root Of 42Square Root Of 25Square Root Of 1681Square Root Of 9375
Square Root Of 11Square Root Of 1521Square Root Of 600Square Root Of 435
Square Root Of 13Square Root Of 300Square Root Of 27Square Root Of 4800
Square Root Of 128Square Root Of 75Square Root Of 5000Square Root Of 6174
Square Root Of 625Square Root Of 35Square Root Of 1200Square Root Of 611
Square Root Of 1000Square Root Of 44Square Root Of 4Square Root Of 1437
Square Root Of 72Square Root Of 82Square Root Of 21Square Root Of 1469
Square Root Of 113Square Root Of 2Square Root Of 124Square Root Of 955
Square Root Of 68Square Root Of 99Square Root Of 2000