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Square Roots – Explanation & Examples

In 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.

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. Evaluate √144 + √196
  2. Simplify √25 x √25
  3. Find 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. Calculate the number total number of seats in a row.
  5. Calculate √5625.
  6. A square garden has an area of 16 square meters. Calculate the perimeter of the garden.
  7. What least number must be added to 570 to make it a perfect square.
  8. Evaluate √0.9 + √2.5.
  9. Find the square root of the first perfect four digit number.
  10. What is √0.0025 ?

 

Answers to practice questions

1. √144 + √196

= 12 + 13

= 25

2. √25 x √25

= 5 x 5

= 25

3. √1000000

1000000 has an even number of zero, therefore pick each zero from a pair.

= 1000

4. Equal number of rows and columns

Number of seats in a row and column = √ 3136

56 seats

5. √5625

= 75

6. √16 = 4

Perimeter = 4 x 4

= 16 meters

7. 570 + 6 = 576

√576 = 24

8. √0.9 + √2.5

= 0.3 + 0. 5

= 0.8

9. The first perfect four-digit number is 1024

10. √0.0025

= 0. 05

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