Square Root of 138

‘√’ is the radical symbol used to represent the square root, and the number under the radical symbol is called the radicand. Let ‘n’ be a number and ‘n²’ is the squared number. This means that if ‘n² = m, then √m = n’.

Hence, we can say that the square root is a number that when multiplied by itself results in another number. It can also be expressed in the exponent form, as in (n)^½. The square root of 138 can also be expressed as √138 (radical form) or (138)^½ exponentially. Since the number 138 is not a perfect square, its square root value will be an irrational number. 

For perfect square numbers, we use the prime factorization method. To find the square root of non-perfect squares, we use the long division method and the approximation method. A few commonly used methods for finding square root are as follows:

1. Prime Factorization Method
2. Long Division Method
3. Approximation Method

Prime factorization means breaking a number into its prime numbers and writing it as their product. It is easy to find the square root of a perfect square using prime factorization because you can pair the prime factors. However, for non-perfect squares, finding the square root is more difficult.

Let’s try to find the square root of the number 138 :

Step 1: Find the prime factors of 138. Start dividing it by the smallest prime number, which is ‘2’. If it’s not divisible, move on to the next prime number.

Step 2: Continue the division using the prime numbers until you can’t divide it anymore.

Step 3: The prime factorization of 138 is 2 x 3 x 23

Since there are no repeating factors to make pairs, we cannot find the square root of 138. Thus, 138 is a non-perfect square.

The long division method is a step-by-step process to solve big math problems. It’s useful for finding the square root of both perfect and non-perfect squares. For perfect squares, you get an exact value, while for non-perfect squares, the square root will be an approximate value.

For example, we find the square root of 4 as 2 and the square root of 5 as 2.23. Here, 4 is a perfect square because its square root is a whole number, but 5 is not a perfect square because the square root is a decimal.

Given below are the steps to find the square root of 138 using the Long Division method:

Step 1: Group the digits from left to right. We group 138 as 1 and 38. Add zeros after the decimal to make pairs.

Step 2: Find a number ‘n’ whose square is ≤ 1. We find ‘n’ as 1.

Step 3: Substitute the value of ‘n’ in n² ≤ 1→ 1² ≤ 1 → 1≤ 1. Thus, the quotient is 1 and the remainder is 0

Step 4: Now the new divisor is 2n, where ‘n’ is the previous divisor. Thus, the new divisor is 2 → 2 × n = 2 × 1 = 2

Step 5: Bring down the next pair, 38. So, 38 is the new dividend.

Step 6: Find the digit ‘Y’ where 2Y × Y ≤  38. We find the value of ‘Y’ as 1.

Substituting the value of ‘Y’ in 2Y × Y  → 21 × 1 = 21. Thus, 21 ≤ 38.

Step 7: Now subtract 21 from 38  → 38 - 21 gives 17. Now 17 is the remainder and 11 is the quotient.

Step 8: We will bring down two zeros and the remainder 17 becomes 1700. Now, we double the quotient 11 to get 22. This helps us with our next step in dividing. (11 × 2 = 22).  

Step 9: Find the digit ‘Y’ where 22Y × Y ≤ 1700. We find ‘Y’ as 7.

Substituting the value of ‘Y’ in 22Y × Y → 227 × 7 = 1589. Thus, 1589 ≤ 1700

Step 10: We will repeat steps 7 and 8 until we get the value of the square root of 138 rounded to four decimal places.

The value of √138 ≈ ±11.7473

To find the approximate value of a non-perfect square, we use the closest smaller and larger perfect squares. For example, to get the approximate value of √14, note that it lies between the square roots of the perfect squares 9 (3²) and 16 (4²). So, ​√14 is between 3 and 4.

Follow the steps given below to find the approximate value of non-perfect squares like 138:

Step 1: The square root of 138 lies between the square root of perfect squares 121(11²) and 144 (12²) .

Step 2: Now we will take the smallest perfect square (121), subtract it from both the given number (138) and the greater perfect square (144), and then divide the subtracted values.

→ (138 - 121) ÷ (144 - 121) =0.739 ≈  0.74

Now add 11 to 0.74 to get the approximate square root value of 138.

Since the square root of 121 is closer to and smaller than 138, the initial digits of the square root of 138 will be 11.

→ 11 + 0.74 = 11.74

So, √138 ≈ ±11.74

• Divisible: A number is said to be divisible if the given number completely divides another, which leaves ‘0’ as the remainder. For example, 45 is divisible by 9. Here the quotient is 5 and the remainder is 0.

• Non-perfect Square: Any number that is not possible to write as the product of an integer which gets multiplied by itself. For example, 8 is not a perfect square because there is no number that when multiplied by itself gives 8 as the product.

• Dividend: Any number that is being divided. For example, in 65 ÷ 5, 65 is the dividend.

• Divisor: Any number that divides the dividend. For example, in 56 ÷ 8, 8 is the divisor.

• Remainder: Number that is left behind after the division process. For example, in 60 ÷ 7, we get 4 as the remainder and 8 as the quotient.

• Prime Factorization: The process of breaking down a number into its prime factors. For example, the prime factors of 30 are 2, 3, and 5.

• Irrational Number: A number that cannot be expressed as p/q, where q  ≠ 0. For example, decimal numbers like  3.4, 98.2, 134.6, and so on are irrational.

• Radical Symbol: A symbol that denotes the square root of a number. For example, the square root of 300 is denoted as √300, where ‘√’ is the radical symbol.