Square Root of 69

To find the square root of 69, we can use any method – prime factorisation, Newton’s formula for square, long division method, etc. As 69 is not a perfect square number, we get an irrational root. A perfect square number is a number that can be written as a square of an integer and whose square is a whole number. The square root of a number is just the opposite of squaring. In radical form, the square root of 69 is denoted as √69, and in the exponential form, it is written as (69)^½. Let us learn how to find the square root of 69.

Learn more about square and square roots.

What Is the Square Root of 69?

The square root of 69 is a number whose square gives the result 69. Now, 69 is not a perfect square number because we cannot find any integer which could be multiplied twice to get 69. Thus, we find an approximate value of the square root of 69, as it is an irrational number.

The square root of 69 can also be determined by finding the roots of the quadratic equation

x² – 69 = 0

⇒ x² = 69

Taking square roots on both sides, we get,

⇒ x = √69

⇒ x = ± 8.30662

How to Find the Square Root of 69?

Let us find the square root of 69 using Newton’s square root equation, prime factorisation, and the long division method. The repeated subtraction method will not work, as 69 is not a perfect square, and its square root is an irrational number.

Newton’s Square Root Equation

Newton’s square root equation to find the square root of any number is given by

√N ≈ ½ (N/A + A) where,

N is the number whose square root we need to find.

A is the approximate guess square root number.

Now, we have N = 69, as 8² = 64 < 69, we take A = 8. Let us put all the values in the given formula to find the square root of 69.

√69 ≈ ½ (69/8 + 8) = ½ (8.625 + 8) ≈ 8.31, which is quite close to the actual square root of 69.

Prime Factorisation Method

To find the square root of 69 by the prime factorisation method, we first prime factorise 69 and then make pairs of two to get the square root. Since 69 is a prime number,

Prime factorisation of 69 = 13 × 3

Square root of 69 = √[13 × 3] = √69

Long Division Method

To find the square root of 69 by the long division method, we shall write 69 as the dividend and pair its digits from right to left. Now, we shall calculate the square root:

To learn how to find the square root of any number by the long division method, click here.

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Solved Examples on Square Root of 69

Example 1:

Find the semi-perimeter of the circle whose area is 69𝜋 m².

Solution:

Let r be the radius of the circle.

The area of the circle = 𝜋r² = 69𝜋 m²

⇒ r² = 69 (taking square root on both sides)

⇒ r = √69

⇒ r ≈ 8.306 m (taking the positive root as length cannot be negative)

∴ the semi-perimeter of the circle = 𝜋 × 8.306 = 26.08 m (approx.)

Example 2:

Find the length of the base of a triangle whose area is 6.9 m² and the height is one-fifth of its base.

Solution:

Let b be the base of the triangle.

Height of the triangle = b/5

Area of the triangle = ½ × b × b/5 = 6.9 m²

⇒ 1/10 × b² = 6.9

⇒ b² = 6.9 × 10 = 69 (taking square root on both the sides)

⇒ b = √69 ≈ 8.306 m

∴ the length of the base of the triangle is 8.306 m.

Example 3:

What is the smallest number that should be subtracted from 69 to make it a perfect square number? Also, find the square root of that number obtained.

Solution:

Clearly, 8² = 64 < 69

Therefore, we must subtract 5 from 69 to make it a perfect square number.

√64 = ± 8