Square Root Formula Steps And Solved Examples
The square root of a number x is the value which when multiplied by itself gives the original number x as its product. The square root of a number x is denoted by \[\sqrt{x}\].
Mathematically, if y is the square root of a number x, i.e., \[\sqrt{x}\] = y, then multiplying y with itself gives x as its product, i.e., y × y = \[y^2\] = x.
For example, the square root of 4 = \[\sqrt{4}\] = 2, the square root of 25 = \[\sqrt{25}\] = 5, etc.
In this article, you will learn different methods to find the square root of a given number.
There are two methods which are generally used for finding the square root of a number. These two methods are:
(1) Prime Factorisation Method
(2) Long-Division Method
METHOD 1
Finding Square Root of a Perfect square number by Prime Factorisation Method
When a given number is a perfect square, the following steps are used to find its square root:
Step 1: Resolve the given number into its prime factors.
Step 2: Make pairs of similar prime factors.
Step 3: Separate out one factor from every pair.
Step 4: Multiply these separate out factors, their product will give the required square root.
For Example: Find the square root of 256 by Prime factorisation method.
We will follow the above steps to find the square root of 256:
Step 1: Resolve the given number 256 into its prime factors.
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So, 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
Step 2: Make pairs of similar prime factors.
256 = \[\underline{2 × 2}\] × \[\underline{2 × 2}\] × \[\underline{2 × 2}\] × \[\underline{2 × 2}\]
Step 3: Separate out one factor from every pair.
Step 4: Multiplying these separate out factors, we get their product as; 2 × 2 × 2 × 2 = 16
Therefore, the square root of 256 = \[\sqrt{256}\] = 2 × 2 × 2 × 2 = 16.
METHOD 2
Finding Square Root of a Perfect square number by Long Division Method
To learn the long division method for finding the square root of a number, let us consider an example, Find the square root of 529 by long division method.
Step 1: Place a bar over every pair of digits starting from the digit at the unit's place. If the number of digits in the given number is odd, then the left-most single digit too will have a bar. Thus we have, \[\overline{5}\] \[\overline{29}\].
Step 2: Find the largest number whose square is less than or equal to the number under the left-most bar (22 < 5 < 32 ). Take this number as the divisor and the number under the extreme left bar as the dividend (here 5) for division and get the remainder.
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Step 3: Write down the number under the next bar, which is 29 in this case, to the right of the remainder. So the new dividend is 129.
Step 4: Double the quotient and write it with a blank on its right.
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Step 5: Choose a largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product obtained is less than or equal to the dividend. In this case 42 × 2 = 84. As 43 × 3 = 129. So we choose the new digit as 3. Perform the division with new divisor 43 and dividend 129 to get the remainder.
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Step 6: Since the remainder obtained is 0 and no digits are left in the given number, therefore, the square root of 529 = \[\sqrt{529}\] = 23.
How to Solve the Square Root Equations?
Let us consider an example to learn how to solve the square root equations. Consider a square root equation below and find the value of x:
\[\sqrt{x-2}\] = 5.
To solve the above equation, we will first remove the square root from LHS of the equation by squaring both LHS and RHS.
\[(\sqrt{x-2})^2\] = (5)2
⇒ x - 2 = 25
⇒ x = 25 + 2
⇒ x = 27
Therefore, the required value of x for the given square root equation is 27.
FAQs on How To Calculate The Square Root Of A Number
1. What is the square root of a number?
The square root of a number is a value that, when multiplied by itself, gives the original number. In other words, if a × a = b, then a is the square root of b.
- Example: Since 5 × 5 = 25, the square root of 25 is 5.
- Mathematically written as: √25 = 5.
- Every positive number has two square roots: one positive and one negative (±).
2. How do you find the square root of a number?
You can find the square root of a number using prime factorization, long division method, or a calculator. The method depends on whether the number is a perfect square.
- Prime Factorization: Break the number into prime factors and pair them.
- Long Division Method: Used for large or non-perfect squares.
- Calculator Method: Press the √ button.
- Example: √36 = 6 because 6 × 6 = 36.
3. What is the formula for square root?
The square root formula is written as √x = x1/2. This means that the square root of a number is the same as raising it to the power of one-half.
- Example: √16 = 161/2 = 4.
- If x² = y, then √y = x.
4. How do you find the square root using prime factorization?
To find the square root using prime factorization, factor the number into primes and group them in pairs.
- Step 1: Prime factorize the number.
- Step 2: Make pairs of identical factors.
- Step 3: Take one number from each pair and multiply them.
- Example: 36 = 2 × 2 × 3 × 3 → √36 = 2 × 3 = 6.
5. What is the square root of a negative number?
The square root of a negative number is not a real number but an imaginary number. It is written using i, where i = √−1.
- Example: √(−9) = 3i.
- Negative numbers do not have real square roots.
6. What is a perfect square?
A perfect square is a number that is the square of an integer. Its square root is always a whole number.
- Examples: 1, 4, 9, 16, 25.
- √49 = 7 because 7 × 7 = 49.
7. How do you find the square root by the long division method?
The long division method finds the square root of large or non-perfect square numbers step by step by grouping digits in pairs.
- Step 1: Pair digits from right to left.
- Step 2: Find the largest square less than the first pair.
- Step 3: Subtract and bring down the next pair.
- Step 4: Repeat the process.
- This method works for both perfect and non-perfect squares.
8. What is the difference between square and square root?
The square of a number is the result of multiplying it by itself, while the square root is the number that produces the original number when squared.
- Square: 6² = 36.
- Square Root: √36 = 6.
- Square is power 2; square root is power 1/2.
9. Can a number have two square roots?
Yes, every positive number has two square roots: one positive and one negative. These are called positive and negative square roots.
- Example: √25 = ±5.
- However, the symbol √25 usually represents the principal (positive) square root, which is 5.
10. How do you estimate the square root of a non-perfect square?
You can estimate the square root of a non-perfect square by identifying the two closest perfect squares.
- Example: To estimate √20.
- 16 < 20 < 25.
- √16 = 4 and √25 = 5.
- So √20 lies between 4 and 5, approximately 4.47.
