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Square Root of 76

76 has more than two factors. It has 2 and 19 as its prime factors and the sum of the prime factors is 21. In this lesson, we will learn to calculate the square root of 76 by long division method. We will also go through a few solved examples and interactive questions related to the square root of 76.

Square Root of 76: √76 = 8.717
Square of 76: 76² = 5,776

1.	What Is the Square Root of 76?
2.	Is Square Root of 76 Rational or Irrational?
3.	Tips and Tricks
4.	How to Find the Square Root of 76?
5.	FAQs on Square Root of 76

What Is the Square Root of 76?

Square root of 76 is the number that when multiplied by itself results in the original number. As there is no such integer which on multiplying with itself gives 76 exactly, the square root of 76 is not a whole number.

Is the Square Root of 76 Rational or Irrational?

√76 = 8.71779788 is a non-recurring and non-terminating decimal number. On squaring 8.71779788, we get (8.71779788)² = 75.999999. This value is approximately 76. This shows that 76 is not a perfect square which also proves that the square root of 76 is an irrational number.

Tips and Tricks:

76 is not a perfect square as the square root of 76 is an irrational number. The square root of any number n, which is not a perfect square, will always be an irrational number.

How to Find the Square Root of 76?

To find the square root of 76, we can use the following steps:

Step 1: First, we will check whether the number is a perfect square or not. 76 is not a perfect square as it cannot be expressed as the square of a number.
Step 2: Once the number is checked, we can do the following:

A number which is a perfect square can be written as √x² = x.
For a number which is not a perfect square, the square root can be found using the long division method. This number can be written in its simplified radical form of the square root.

76 is not a perfect square. Hence, we will find the square root of 76 using the long division method. The simplified radical form of the square root of 76 is given below.

Simplified Radical Form of Square Root of 76

76 can be written as the product of 2 and 38. It is given as:

√76 = √(2 × 38)

2 is not a perfect square. Hence, it stays within the root sign. 38 can be shown as 2 × 19. The simplified radical form of the square root of 76 is 2√19.

Square Root of 76 by Long Division Method

The square root of 76 can be found using the long division method. These are the steps to be followed:

Step 1: We start pairing the digits from the right. We will pair up the digits by placing a bar above them.
Step 2: We need to find a number such that when it is multiplied with itself, the product is less than or equal to 76. Keeping the divisor as 8, we get the quotient as 8 and the remainder 76-64 = 12.
Step 3: Double the divisor and enter it with a blank on its right. Then assume the largest digit to replace the blank; this will become the new digit in the quotient. Now, when the new divisor will be multiplied to the new quotient, the final product will be lesser than or equal to our dividend. Finally, divide and write the remainder. Bring down a pair of zeros to form the new dividend. Repeat this process to get the decimal places you want.

Hence, √76 = 8.717

Explore square roots using illustrations and interactive examples

Challenging Questions:

How will Justin find the square root of 76 using the long division method up to 8 decimal places?
How will Shawn express the square root of 38 in terms of the square root of 76?

Example 1: Evaluate (2√19 + √76) × (4√19 - √76)

Solution

√76 can be simplified as 2√19
Hence, (2√19 + √76) = (2√19 + 2√19) = 4√19
(4√19 - √76) = (4√19 - 2√19) = 2√19
For finding (2√19 + √76) × (4√19 - √76), we take the product of the above obtained answers which gives (2√19 + √76) × (4√19 - √76) = 4√19 × 2√19 = 8 × 19 = 152.
Example 2: What is the circumference of a circle if its area is 76π square inches?

Solution

The area of a circle is given as πr².
Hence, area = πr² = 76π ⇒ r² = 76 ⇒ r = √76 = 8.717 ≈ 8.7 inches.
Circumference of circle = 2πr = 2 × π × 8.7 = 54.63 inches. Thus, the circumference of the circle is 54.66 inches.

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FAQs on Square Root of 76

What is the square root of 76 in the simplest form?

The square root of 76 in the simplest form is expressed as 2√19.

How do you find the square root of 76?

As 76 is not a perfect square number, its square root can be obtained using the long division method.

What is the approximate square root of 76?

The square root of 76 is 8.71779788 (approximately).

Is the square root of 76 rational or irrational?

The square root of 76 is irrational.

Is the square root of 76 a real number?

Yes, the square root of 76 is a real number.

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