Square Root of 75

In Mathematics, the square root of a number is a number, which when multiplied by itself results in the original number. Thus, the square root of 75 is a number, which when multiplied by itself results in the original number 75. The square root of 75 is not a rational number since the value of square root 75 cannot be expressed in the form of p/q. The square root of 75 can be found using two different methods, such as long division and prime factorization methods. In this article, we are going to learn the square root of 75 in decimal form, radical form and the procedure to find the value of square root 75 using the two different methods with complete explanation.

Table of Contents:

• What Is The Square Root Of 75?
• Square Root Of 75 In Radical Form
• Square Root Of 75 By Prime Factorization Method
• Square Root Of 75 By Long Division Method
• Examples
• FAQs

What is the Square Root of 75?

If a number is multiplied by itself and gives the result as 75, then the number is the square root 75. The square root of 75 is symbolically expressed as √75.

Hence, √75 = √(Number × Number)

Thus, if we multiply the number 8.66 two times, we get the original value 75.

(i.e) √75 = √(8.66×8.66)

√75 = √(8.66)²

Now, remove square and square root, we get

√75 = ± 8.660 (or)

√75 = ± 8.66 (Rounded to two decimal places)

Square Root of 75 in Radical Form

We can also represent the square root of 75 in the radical form. The radical form of the value of the square root of 75 can be written if we know the prime factorisation of 75. We know that the prime factorisation of 75 is 3×5×5. Therefore, the simplest radical form of value of the square root of 75 is √3×√5×√5, which is equal to 5√3

Square Root of 75 by Prime Factorization Method

To find the square root of 75 using the prime factorization method, we need to know the prime factorisation of 75. The prime factorisation of 75 is 3×5×5.

Thus, √75 = √3. √5.√5 = 5√3.

We know that,

√3 = 1.732

Now, substitute the values of √3 in the above equation, we get

√75 = 5 × 1.732

√75 = 8.660 (approximately)

Hence, the square root of 75 in decimal form is 8.66 (rounded to two decimal places)

Square Root of 75 by Long Division Method

The procedure to find the square root of 75 using the long division method is given as follows:

Step 1: Write the number 75 in decimal form. To find the exact value of the square root of 75, add 6 zeros after the decimal point. Hence, 75 in decimal form is 75.000000. Now, pair the number 75 from right to left by putting the bar on the top of the number.

Step 2: Now, divide the number 75 by a number, such that the product of the same number should be less than or equal to 75. Thus, 8×8 = 64, which is less than 75. Thus, we obtained the quotient = 8 and remainder = 11.

Step 3: Double the quotient value, so we get 16, and assume that 160 is the new divisor. Now, bring down the value 00 for division operation. So, the new dividend obtained is 1100. Now, find the number, such that (160 + new number) × new number should give the product value, that should be less than or equal to 1100. Hence, (160+6) × 6 = 996, which is less than 1100.

Step 4: Now subtract 996 from 1100, and we get 104 as the new remainder and 86 as a quotient.

Step 5: The new quotient obtained is 86, and double that. Hence, we get 172 and assume that 1720 is our new divisor. Now, bring down the two zeros and, we have 10400 as the new dividend.

Step 6: Find the number, such that (1720 + new number) × new number should give the product value, that should be less than or equal to 10400. Thus, (1720+6)× 6 = 10356, which is less than 10400.

Step 7: Subtract 10356 from 10400, and we get 44 as the new reminder.

Step 8: Continue this process until we get the approximate value of the square root of 75 up to three decimal places. (Note: keep the decimal point in the quotient value after bringing down all the values in the dividend).

Step 9: Thus, the approximate value of the square root of 75, √75 is 8.66 (rounded to 2 decimal places).

Video Lessons on Square Roots

Visualising square roots

Finding Square roots

Examples

Example 1:

Simplify 20 +√75.

Solution:

Given: 20 +√75.

We know that the square root of 75 is 8.66.

Now, substitute the value in the expression, we get:

20 +√75 = 20+8.66

20 +√75 = 28.66

Therefore, 20+√75 is 28.66.

Example 2:

Find the value of a, if 10a + √75 = 45

Solution:

Given equation: 10a + √75 = 45…(1)

We know that √75 =8.66.

Now, substitute the value in equation (1),

10a + 8.66 = 45

10a = 45-8.66

10a = 36.34

a = (36.34)/10

a = 3.634

Therefore, the value of a is 3.634.