Squares and Square Roots Examples
Example 1: Find the square of 23.
Solution:
232 = (20 + 3)2
= 20(20 + 3) + 3(20 +3)
= 202 + 20 × 3 + 3 × 20 + 32
= 400 + 60 + 60 + 9
= 529
Example 2: Find the square of 39.
Solution:
392 = (30 + 9)2
⇒ 392 = 30 (30 + 9) + 9 (30 + 9)
⇒ 392 = 302 + 30 × 9 + 9 × 30 + 92
⇒ 392 = 900 + 270 + 270 + 81
⇒ 392 = 1521
Example 3: Find the square root of 144.
Solution:
144 = (2 × 2) × (2 × 2) × (3 × 3)
⇒ 144 = 22 × 22 × 32
⇒ 144 = (2 × 2 × 3)2
⇒ 144 = 122
Therefore, √144 = 12
Sometimes a number is not a perfect square.
Example 4: Is 2352 a perfect square? If not, find the smallest multiple of 2352 which is a perfect square. Find the square root of the new number.
Solution:
2352 = 2 × 2 × 2 × 2 × 3 × 7 × 7
⇒ 2352 = 24 × 3 × 72
As the prime factor 3 has no pair, 2352 is not a perfect square.
If 3 gets a pair then the number will become perfect square.
2352 = 2 × 2 × 2 × 2 × 3 × 7 × 7
⇒ 2352 = 24 × 32 × 72
⇒ 2352 = 22 × 3 × 7 = 84
Example 5: Square root of 19.36
Solution:
Step 1: Make pairs of an integral part and decimal part of the number. Add a zero to the extreme right of the decimal part if required.
Step 2: Find the perfect square of an integral part, find the number closest to the integral part (Either small or equal). In this case, the square of 4 is 16 which is closest to 19:
Step 3: Put the decimal Part next to the Remainder obtained. Double the divisor of an Integral Part and place it in the next divisor, now we have to find the unit place value of this number.
Step 4: Now we have to find the unit place’s number which should be multiplied in order to get 336, here we can see, if we multiply 84 with 4, we will get 336.
Hence, we obtained 4.4 as the square root of 19.36
Squares and Square Roots
Squares and Square roots are highly used mathematical concepts which are used for various purposes. Squares are numbers produced by multiplying a number by itself. Conversely, the square root of a number is the value that, when multiplied by itself, results in the original number. Thus, squaring and taking the square root are inverse operations.
This can be understood with the help of an example such as take q number 3 then its square is 32 = 9. Now the square root of 9 is √(9) = 3. Thus, it is evident that the square root is the inverse operation of the square.
Let’s learn more about square and square roots in this article including properties of square numbers, squares of different types of numbers, properties of square roots, etc.
Table of Content
- What is a Square of a Number?
- Square and Square Roots Table
- Representation of Square
- Squares of Negative Numbers
- Square of 2
- Properties of Square Numbers
- Square Numbers 1 to 30
- Numbers Between Squares
- What are Square Roots?
- Representation of Square Root
- Properties of Square Root
- Square Roots of Perfect Squares
- Square Root of Imperfect Squares
- Square Root of Numbers 1 to 30
- Square Roots 1 to 50 – Table
- Finding Square Roots
- Finding the square root of decimal numbers
- Interesting Patterns in Square Roots and Squares
- Adding Triangular Numbers
- Squares of 1, 11, 111, 1111…
- Squares of Numbers with 5 as a Unit Digit
- Squares and Square Roots Examples
- Square and Square Roots Class 8
- Applications of Square and Square Roots