Squares and Square Roots

Square Root of Perfect and Non Perfect Squares

Till now we are quite familiar with the terms square numbers, perfect squares, and square roots. Also, we have discussed the basic properties of perfect squares and that of square roots. We are also quite familiar with the methods of calculating squares and checking for perfect square and estimating square root. In this section, we will discuss some methods of calculating square roots for perfect and Non-perfect squares.

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Methods of Estimating Square Roots of Perfect Squares and Non-Perfect Squares

Let us discuss some of the methods for calculating square root for perfect squares. The same method can be applied to non-perfect squares too.

Prime Factorization Method

As the name suggests in this method we need to find the prime factors of the perfect square. The perfect square is factorized into its prime factors by successive division. The pairs of the prime factors are paired. Taking the product of one factor from each pair will result in the square root of the perfect square.

  • Let us find the square root of 441. The prime factorization of 441 is 441 = 3 × 3 × 7 × 7. Pairing the prime factors and selecting one from each pair gives 3 × 7 = 21. So, the square root of 441 = √441 = 21.
  • Let us find the square root of 180. The prime factorization of 180 is 180 = 2 × 2 × 3 × 3 × 5. If we make the pair of the prime factors we see that the prime factor 5 is not in the pair. Therefore 180 is not a perfect square and hence the square root is not perfect. The square root of 180, √180 = 2 × 3 × √5 = 6√5.

Division Method

Calculating a square root for a large number by prime factorization method often time-consuming. To overcome this, we use the division method to find the square root of a large number. Steps for calculating square root by division method for perfect squares

  • Take the number whose square root is to find.
  • Place a bar over every pair of the digit of the number starting from that in unit’s place (rightmost side).
  • We divide the leftmost number by the largest number whose square is less than or equal to the number under the leftmost bar.
  • Take this number as the divisor and the quotient. The number under the leftmost bar is considered to be the dividend.
  • Divide and get the remainder.
  • Bring down the number under the next bar to the right of the remainder.
  • Double the divisor (or add divisor to itself).
  • To the right of this divisor find a suitable number which together with divisor forms a new divisor for the new dividend. The new number in the quotient will have the same number as selected in the divisor. The condition is the same as being either less or equal to that of the dividend.

This process continues till we get zero as the remainder. The quotient thus obtained will be the square root of the number.

Example for Division Method

Let us find the square root of 225. Placing bar over the pair of digits in 225 starting from the unit place.

Start the division from the leftmost side. Here 1 is the number whose square is less than 2. Putting it in the divisor and the quotient and then doubling it will give

Perfect Squares

Now we need to find a number for the blanks in divisor and quotient. Let that number be x. We need to check when 2x multiplies by x gives a number which is either less than or equal to 125. Take x = 1, 2, 3 and so on and check.

In this case, 24 × 4 = 96 and 25 × 5 = 125. So we choose x = 5 as the new digit to be put in divisor and in the quotient. The remainder here is 0 and hence 15 is the square root of 225. If the number is not a perfect square, the remainder obtained is not zero. It will always leave some remainder other than zero.

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Square Roots of Decimals

Finding the square root of decimals is similar to that of numbers with no decimals. The only difference is that in the manner of putting a bar over the numbers before and after the decimal point. We place the bar over every pair of the number starting from the unit’s place (before the decimal point).  And we place the bar over every pair of the number starting from the beginning of the number after the decimal point.

If only one digit is there after the decimal point we can add zero to the right of the number (after the decimal point) and make the pair. Consider finding the square root of 132.25. The bar is placed over 32 as a pair and 25 as another one. In 156.351 the bar is placed over 56 and after the decimal point, the bar is placed over 35 and 10 (we add zero to the right of one).

Solved Examples for You

Question 1: If the perimeter of a rectangle is 196 cm and the breadth is 4 less than that of its length. The area of a square is 196 sq. cm. Compare the lengths of both the rectangle and the square.

Answer : Perimeter of a rectangle is 2(l + b) = 196.
or, l + b = 98. The breadth is 4 less than the length i.e, b = l – 4.
We have, l + l – 4 = 2l – 4 = 98 or, l = 51 and b = 47.
Area of the square is 196 sq. cm. Then the length of the square = 2 = 196 or, l = 14.
Here we see that the length of the square is smaller than that of the rectangle.

Question 2: Is 1253.56 a perfect square?

Answer : Using the Division method for finding the square root, we have

Perfect Squares

Since, we are getting a remainder other than zero (i.e., 40). So, 1253.56 is not a perfect square.

Question 3: Is 0 a perfect square?

Answer: 0 is a perfect square. As you know, a perfect square refers to a number whose roots are a rational number. Thus, because 0 is a rational number as we can express it as 0/1. Thus, it brings us to our answer that 0 is a perfect square.

Question 4: What is the perfect square trinomial?

Answer: A trinomial is a perfect square trinomial if we can factor it into a binomial that multiplies to itself. In a perfect square trinomial, two of the terms are going to be perfect squares. For instance, we take the trinomial x2 – 12x + 36. We see that both x2 and 36 are perfect squares

Question 5: How many perfect squares are there between 1 and 20?

Answer: As you all know, a perfect square refers to a number where its square root is an integer. Thus, it brings us to that between 1 and 20, there are four perfect squares. They are 1, 4, 9, and 16.

Question 6: Is 1000 a perfect square?

Answer: A number will be a perfect square or a square number if its square root is going to be an integer. To be exact, it is the product of an integer with itself. So, when we look here, we see the square root of 1000 is around 31.623. Therefore, the square root of 1000 will not be an integer, and thus, 1000 is not a square number.

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