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In this lesson, we will study square roots. Square root is exactly opposite of square. We will study three different methods of square root.
- In the square there is multiplication and in the square root there is division.
- All different methods to find square roots are based on divisions.
- To study square roots, we have to remember the squares of 1 to 10 numbers.
- The symbol of square root is√.
| Table of Content |
Key Takeaways: Square, Square root, Multiplication, Division
Also read: Isosceles Triangle Theorems
What are Square Roots
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The square root of any number is equal to a number when squared gives the original number.
X=X2=Y=√Y=X
2= 22= 4= √4=2
To find the square root it is necessary to identify whether a number is perfect square or imperfect square. If the number is 4,9,16, 25 etc these are perfect square numbers and if the numbers are 2,3,5,6 etc these are imperfect squares numbers.
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Methods to find Square Roots
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The following methods to find square roots are:
Square root by long division method
This method is suitable to get the square root of the bigger value number by creating the pairs of that given number first. To study this method it is necessary to follow the example given below and their steps carefully.
Square root by prime factorization method
In this method, take the Least common multiple (LCM) of a given number, and it is necessary to find the prime factors. It is a simple method to get a square root.
Square root of decimal number
It is similar to the first method i.e. the long division method. In this method we find the square root of decimal numbers like 3.6, 6.4, 2.56 etc. In this method proper pairing is important. It is a very important step to pair the decimal number, for pairing on the right side of point is right to left and pairing for left side of point is from left to right.
Square root of estimation method
In this method we need to remember the square roots of perfect numbers like 2,3,4,5,6 etc. For example, to estimate the square root of 30, As we know the square of 5 is 25 and 6 is 36, that means the square root of 30 lies between 5 and 6. It will be in decimal numbers. Similarly, the square root of 110 lies between 10 and 11.
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To solve the square root equation
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The following steps are to solve the square root equation:
- Isolate the squares first.
- Squaring the both sides i.e. LHS (left hand side) and RHS (right hand sides).
- Solve the rest of the equation.
Squares and squares root
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Square and square roots of perfect square are as follows:
| Number (x) | Squares (x2) | Number (x) | Square root (√x) |
|---|---|---|---|
| 2 | 4 | 4 | 2 |
| 5 | 25 | 25 | 5 |
| 6 | 36 | 36 | 6 |
| 7 | 49 | 49 | 7 |
| 8 | 64 | 64 | 8 |
| 9 | 81 | 81 | 9 |
| 10 | 100 | 100 | 10 |
| 11 | 121 | 121 | 11 |
| 12 | 144 | 144 | 12 |
| 13 | 169 | 169 | 13 |
| 14 | 196 | 196 | 14 |
| 15 | 225 | 225 | 15 |
| 16 | 256 | 256 | 16 |
| 17 | 289 | 289 | 17 |
| 18 | 324 | 324 | 18 |
Square roots of non squares number (Imperfect squares) are as follows:
| Square root √x | x |
|---|---|
| √2 | 1.414 |
| √3 | 1.732 |
| √5 | 2.236 |
| √6 | 2.449 |
| √7 | 2.645 |
| √8 | 2.828 |
| √11 | 3.316 |
Points to Remember
Following are some important points:
- √ 2.56=1.6
- √ 256=16
- Pairing in decimal type problems (follow the arrow given):
- 744
← →
- 74 4
7 are the first pair on the left, and 74 are the first pair on the right side and 4 are the second pair on the left side.
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Sample Questions
Ques: Find the square root 54756 by long division method. (6 Marks)
Ans: Following are the steps:
Step 1: Make the pairs of the given number; the pair must be from right to left
5 47 56
← ← ← Pairing from right to left
Three pairs 5, 47, 56
Step 2: Take a first square number i.e. 5 and find the square of less than 5 (5 47 46)
12 = 1
22 = 4
Step 3: Multiply the numerator by 2
Numerator = 2
2×2 = 4
Find the number in the blank space whose multiplication value would be equal to or less than 147, if we take 4, then 44 ×4 = 176 which is more than 147. That is why we selected 3. Write this 3 in numerator and also near 4 that will be 43
43 ×3 = 129
Step 4: As per rule we have written 3 in numerator and near 4
147- 129 =18
Bring the 56 down and we get 1856
Step 5: Multiply the numerator by 2
23×2 = 46
Step 6: Find the number whose multiplication value equal to or less than 1856
464 × 4 = 1856
Write the 4 in the numerator and we get 234, Also write the number 4 near 46 and it would be 464. And we get 0 in the remainder.
Answer = value in the numerator
Answer = 234
√ 54756 = 234
Ques: Find the square root of 1764 by prime factorization method. (5 Marks)
Ans: Following are the steps:
Step 1: Take the LCM of 1764
We take LCM until we get the value 1
1764 = 2 × 2 × 3 × 3 × 7 × 7
= (2 × 2) × (3 × 3) × (7 × 7)
Step 2: Make the pair of same number and select one
1764 = 2 × 2 × 3 × 3 × 7 × 7
= (2 × 2) × (3 × 3) × (7 × 7)
2, 3, 7
Multiply the common number from the pair
2×3×7 = 42
√1764= 42
Ques: Find the square root of 2.56. (5 Marks)
Ans: Following are the steps:
Step1: Pair the number.
It is a very important step to pair the decimal number, for pairing on the right side of point is right to left and pairing for left side of point is from left to right.
2 . 5 6 56 is one pair and 2 is another pair
← →
Suppose we take 12.752,
Pairing would be 12. 75 2 — On right side- 75 is first pair and 2 is second pair
On the left side- 12 is the pair
Step 2: Take the least square number which is less than 2.
Least square number less than 2 is 1.
Step 3: Multiply the numerator by 2.
1×2 = 2
Also, (.) point comes to denominator also same time point comes in numerator, it would be:
Step 4: Take the number in the blank space whose multiplication value would be 156 or less than 156
2 6 × 6 = 156
The answer is 1.6
Ques: Solve the radical equation: √(x+2) + 4 = x. (3 Marks)
Ans: Isolate the squares and squaring both sides.
(√(x+2))2 = (x-4)2
x + 2 = x2 – 8x + 16
0 = x2 – 9x + 14
0 = (x-2) (x-7)
x=2, x=7
Ques: Find the square root 5 by estimation method. (3 Marks)
Ans: The square of 2 is 4 and 3 is 9, it means the square root lies between 2 and 3.
2.22 = 4.84
2.82= 7.84
Since we can estimate the square root of 5 is 2.2 approximately.
Ques: What is the square root of non perfect square numbers 2, 3, 5? (2 Marks)
Ans: Square root of non perfect square numbers 2, 3, 5 are given below:
| √2 | 1.414 |
| √3 | 1.732 |
| √5 | 2.236 |
Ques: Find the square root of perfect squares 11 to 15. (2 Marks)
Ans: Square root of perfect squares 11 to 15 are given below:
| 121 | 11 |
| 144 | 12 |
| 169 | 13 |
| 196 | 14 |
| 225 | 15 |
Also read:
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