Square Root of 72

Square root of 72 is equal to 6√2 = 8.48528137424…, i.e. √72 = 8.48528137424… Let’s recall the meaning of square root and the methods used to estimate the value of the square root of 72. When we multiply the value of the square root of a number by itself, it produces the original number. Therefore, the square root is the inverse or reverse process of squaring a number. In this article, you will learn the meaning of the square root of 72 and the value of the square root of 72 in decimal form, radical form, exponential form, surd form, along with the different methods to compute the square root of 72, using step by step procedures.

What is the Square Root of 72?

The square root of 72 is the number which when multiplied by itself gives the original number, i.e. 72. That means, if √72 = m, then m × m = m² = 72. The square root of 72 is denoted by √72, where ‘√’ is the radical symbol and 72 is the radicand. This can be read as “root 72”.

Square Root of 72 Rational or Irrational

The square root of 72 is not a whole number and hence, the number 72 is not a perfect square number. The surd form of the square root of 72 is 6√2, where √2 is an irrational number and thus, √72 is also irrational. Also, the decimal expansion of the square root of 72 is non-repeating and non terminating, so it is an irrational number.

Facts about the Square Root of 72

How to Find the Square Root of 72?

The methods used to find the square root of 72 are:

• Repeated subtraction method
• Prime factorisation method
• Long division method

Let’s learn all these methods of calculating the square root of 72, step by step here.

Square Root of 72 by Repeated Subtraction

In this method, we need to subtract the successive odd numbers, i.e. 1, 3, 5, 7,… till we obtain zero. First, subtract 1 from 72 and 3 from the result obtained in the previous operation, and so on. The number of odd numbers we subtract in this process is treated as the square root of 72.

• 72 – 1 = 71
• 71 – 3 = 68
• 68 – 5 = 63
• 63 – 7 = 56
• 56 – 9 = 47
• 47 – 11 = 36
• 36 – 13 = 23
• 23 – 15 = 8
• 8 – 17 = -9

A negative value cannot be the square root of a number; also, we didn’t get 0 in the above subtraction but reached somewhat near after the 8th subtraction. From the above, we can say that the square root of 72 lies between 8 and 9. However, it is to be noted that the square root of non-perfect square numbers cannot be evaluated using the repeated subtraction method.

Learn more about the square root of a number by the repeated subtraction method here.

Square Root of 72 By Prime Factorisation

Step 1: First, we need to write the prime factorisation of 72.

Prime factorisation of 72 = 2 × 2 × 2 × 3 × 3

Step 2: To find the square root, we generally group the same factors in pairs. Let’s group the prime factors.

72 = (2 × 2) × 2 × (3 × 3)

Step 3: Take the square root on both sides.

√72 = √[(2 × 2) × 2 × (3 × 3) = 2 × 3 × √2 = 6√2

Thus, we cannot determine the exact value of √72 using prime factorisation.

Let’s apply the long division method to find the exact value of √72.

Square Root of 72 by Division method

Go through the stepwise procedure given below to find the square root of

Step 1: Put the decimal point and write three pairs of 0s since 72 is a non-perfect square number.

Step 2: Find the largest number whose square is less than or equal to the number in the leftmost group, i.e. 72. We know that 8² < 72 < 9², so we can consider 8. Now, we got 8 as the quotient and 8 as the remainder.

Step 3: Now, bring down 00 and add the divisor with the quotient and enter it with a blank on its right. Thus, 800 will be the new dividend here.

Step 4: Double the divisor, which means 8 + 8 = 16. Write this as one of the digits for the new divisor. Then, identify the largest possible digit, which will also become the new digit in the quotient, so that when the new divisor is multiplied by the new quotient, the product is less than or equal to the dividend. In this case, 164 × 4 = 656, so we choose the new digit as 4. Then, perform the division and get the remainder.

Step 5: Again, bring down the next pair 00 and add the divisor with the quotient and enter it with a blank on its right as we did in the previous step. We can repeat this process as many times as required.

Therefore, the square root of 72 is equal to 8.485…

Square Root of Numbers

Square root of 2 = 1.4142…

Square root of 5 = 2.2361…

Square root of 10 = 3.1623…

Square root of 48 = 6.9282…

Square root of 729 = 27

Get Square Roots of More Numbers here.

Video Lessons on Square Roots

Visualising square roots

Finding Square roots

Solved Examples

Question 1: Simplify 4 square root 72.

Solution:

4 square root 72 = 4 × √72

= 4 × 6√2

= 24√2

= 24 × 1.414 {since root 2 = 1.414 (up to 3 places of decimal point}

= 33.936

Therefore, 4 square root 72 = 4 × √72 = 33.936.

Question 2: Evaluate: √72 + √200

Solution:

Square Root of 72 + Square root of 200 = √72 + √200

= 6√2 + 10√2

= (6 + 10)√2

= 16√2

Or

= 16 × 1.414 {since root 2 = 1.414 (up to 3 places of decimal point}

= 22.624

Therefore, √72 + √200 = 22.624.