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Square Root of 69

The number 69 is a number with only two prime factors 3 and 23. The square root of a number implies a number whose product with itself gives the initial number. We will now find the value of the square root of 69 using various methods and a few interesting facts and problems as well.

Square root of 69: √69 = 8.30662
Square of 69: 69² = 4761

1.	What Is the Square Root of 69?
2.	Is Square Root of 69 Rational or Irrational?
3.	How to Find the Square Root of 69?
4.	FAQs on Square Root of 69
5.	Important Notes on Square Root of 69

What is 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.

Is Square Root of 69 Rational or Irrational?

The square root of 69 is a non-repeating and non-terminating number so, it cannot be expressed in the form of p/q where q ≠ 0.
Hence, the square root of 69 is an irrational number.

How to Find the Square Root of 69?

We will now find the square root of 69 using the below-given methods:

Square Root of 69 Using Approximation Method

Firstly, find two consecutive perfect squares such that 69 lies between them. The two numbers are 64 (8²) and 81 (9²). So, the whole number part of the square root of 69 will be 8.
Now, for determining the decimal part we use the formula: (Given number – Smaller perfect square) / (Greater perfect square – smaller perfect square) = (69 – 64)/(81 – 64) = 5/17 = 0.294
Hence, the approx. the square root of 69 by the approximation method is 8.294.

Square Root of 69 By Long Division

Now we will calculate the square root of 69 by the long division method.

Start pairing the digits from the unit’s place in pairs of two by putting a bar on top of them. We will have one pair in this case (69).
Find a number(a) such that a × a ≤ 69. So, a will be 8 as 8 × 8 = 64.
We get the remainder as 5 (69-64) and the quotient as 8. Now, we will add the divisor a with itself and get the new divisor (16).
Put a decimal in the dividend and quotient simultaneously. Also, place 3 pairs of zero in the dividend part after the decimal.
Bring down one pair of zero. So, our new dividend is 500. Find a number(n) such that 16n × n ≤ 500. The number n will be 3 as 163 × 3 = 489 ≤ 500.
Repeat the above step for the remaining pairs of zero.

So, we get the square root of √69 = 8.306 by the long division method.

Explore square roots using illustrations and interactive examples

Important Notes:

The number 69 is not a perfect square.
The square root of 69 is an irrational number.
The square root of -69 is an imaginary number.

Example 1: Mike wants to find out the square root of -69. Can you help Mike?

Solution:

The square root of all negative numbers is an imaginary number. So, the square of -69 is represented as √-69 = ±8.306i. (where i = √-1).
Example 2: How much minimum area in square feet should be added in a field of surface area 69 square feet, such that it can be converted into a field of square shape with its length as an integer?

Solution:
To convert the field into square-shaped its surface area should be a perfect square. Then the length of the field will be an integer. The nearest perfect square greater than 69 is 81. Therefore, the area of the field that should be added is 81 – 69 = 12 square feet.

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FAQs on Square Roots of 69

What is the negative square root of 69?

The negative square root of 69 is -8.3066.

What is the square root of 69 up to 10 decimal places?

The square root of 69 is √69 = 8.3066238629.

Can we find the square root of 69 using the prime factorization method?

No, we can’t find the square root of 69 via the prime factorization method. Because the number 69 has only two prime factors 3 and 23. So, we can’t reduce it further.

Is the square root of 69 is a rational number?

No, the square root of 69 is not a rational number. Because it cannot be expressed in the form of p/q where q ≠ 0.

Is the number 69 a perfect square?

No, the number 69 is not a perfect square. The square root of 69 is an irrational number and all perfect squares have integers as their square root.

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