Square Root of 69 (How to find) (2024)

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.

Square Root of 69

  • In decimal form: ± 8.30662(approx.)
  • In radical form: ±√69

Square of 69

4761

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:

Square Root of 69 (How to find) (1)

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

Video Lessons on Square Root

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Square Root of 69 (How to find) (2)

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Square Root of 69 (How to find) (3)

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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

Frequently Asked Questions on Square Root of 69

Q1

What is the square root of 69?

The square root of 69 is 8.30662 (approx.).

Q2

Is 69 a perfect square number?

No, 69 is not a perfect square number, as it cannot be expressed as the square of any integer.

Q3

Is the square root of 69 rational or irrational?

The square root of 69 is an irrational number.

Q4

What is the prime factorisation of 69?

The prime factorisation of 69 is 13 × 3.

Q5

Is the square root of 69 a real number?

Yes, the square root of 69 is a real number.

Q6

What is the cube root of 69?

The cube root of 69 is 4.1015 (approx.).

Q7

What number should be subtracted from 69 to get a perfect square number?

As 82 = 64 < 69, 5 must be subtracted from 69 to get a perfect square number.

Square Root of 69 (How to find) (2024)

FAQs

Square Root of 69 (How to find)? ›

The square root of 69 is written as √69 in radical form which is equal to 8.30662 (approximately). The number 69 only has two prime factors that are, 3 and 23. So, its square root cannot be simplified further using prime factorization. The square root of 69 can be written as (69)1/2 in exponential form.

How to solve the square root of 69? ›

The square root of 69 is written as √69 in radical form which is equal to 8.30662 (approximately). The number 69 only has two prime factors that are, 3 and 23. So, its square root cannot be simplified further using prime factorization. The square root of 69 can be written as (69)1/2 in exponential form.

How do you find the answer of a square root? ›

For square roots, find the "reverse" of a square.

When you see a radical, you want to ask yourself, "what number can multiply by itself to give the number under the radical?" For instance, if you see √(9), you want to find the number that can be squared to make nine. In this case, the answer is three, because 32 = 9.

How do you find the square root without guessing? ›

Long division method
  1. Separate your square root base into pairs. ...
  2. Find the largest square that divides into the first number or pair. ...
  3. Subtract the square from the first number or pair. ...
  4. Drop down the next pair. ...
  5. Multiply the first digit of the square by two. ...
  6. Set up the next factor equation.
Aug 8, 2022

Is 69 a perfect square root? ›

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.

What are the factors of 69? ›

Solution: The factors of 69 are 1, 3, 23 and 69 .

What is the square root guessing method? ›

The Babylonian square-root algorithm
  1. Make an initial guess. Guess any positive number x0.
  2. Improve the guess. Apply the formula x1 = (x0 + S / x0) / 2. The number x1 is a better approximation to sqrt(S).
  3. Iterate until convergence. Apply the formula xn+1 = (xn + S / xn) / 2 until the process converges.
May 16, 2016

How did people find square roots without calculators? ›

Just before we had calculators, we used SLIDE RULES. We could use them to multiply, divide, square, square root, sines, cosines, tangents and other stuff, BUT you needed very good eyesight! To find √1.38 we would take out the middle “slide” and go from 1.38 on the A scale and go vertically down to 1.175 on the D scale.

What is the easiest formula for square root? ›

The square root of any number can be expressed using the formula: √y = y½. In other words, if a number has 1/2 as its exponent, it means we need to find the square root of the number.

How to calculate the root of any number? ›

The method relies on the following sequence of steps:
  1. Step 1: Subtract consecutive odd numbers from the number for which we are finding the square root.
  2. Step 2: Repeat step 1 until a value of 0 is attained.
  3. Step 3: The number of times step 1 is repeated is the required square root of the given number.
Oct 2, 2021

Which method is best for square root? ›

If the number is a perfect square, such as 4, 9, 16, etc., then we can factorize the number by prime factorisation method. If the number is an imperfect square, such as 2, 3, 5, etc., then we have to use a long division method to find the root.

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