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Square Root of 69 + Solution With Free Steps

√69 is the expression used to represent the Square root of 69. is the sign used for square root and equals the power of 1/2. The square root of a number is the value, which when taken square of, gives the actual number in answer. this is because the square is canceled by the power 1/2. For example, the square root of 36 is 6.

In this article, we will analyze and find the square root of 69 using various mathematical techniques, such as the approximation method and the long division method.

What Is the Square Root Of 69?

The square root of the number 69 is 8.306.

The square root can be defined as the quantity that can be doubled to produce the square of that similar quantity. In simple words, it can be explained as:

√69 = √(8.306 x 8.306)

√69 = √(8.306)$^2$

√69 = ±8.306

The square can be canceled with the square root as it is equivalent to 1/2; therefore, obtaining 8.306. Hence 8.306 is 69’s square root. The square root generates both positive and negative integers.

How To Calculate the Square Root of 69?

You can calculate the square root of 69 using any of two vastly used techniques in mathematics; one is the Approximation technique, and the other is the Long Division method.

The symbol √ is interpreted as 69 raised to the power 1/2. So any number, when multiplied by itself, produces its square, and when the square root of any squared number is taken, it produces the actual number.

Let us discuss each of them to understand the concepts better.

Square Root by Long Division Method

The process of long division is one of the most common methods used to find the square roots of a given number. It is easy to comprehend and provides more reliable and accurate answers. The long division method reduces a multi-digit number to its equal parts.

Learning how to find the square root of a number is easy with the long division method. All you need are five primary operations- divide, multiply, subtract, bring down or raise, then repeat.

Following are the simple steps that must be followed to find the square root of 69 using the long division method:

Step 1

First, write the given number 69 in the division symbol, as shown in figure 1.

Step 2

Starting from the right side of the number, make the number 69 a pair such as 69.

Step 3

Now divide the digit a by a number, giving a number either 69 or less than 69. Therefore, in this case, the remainder is 5, whereas the quotient is 8. If the number is not a perfect square, add pair of zeros to the right of the number before starting division.

Step 4

After this, bring down the next pair 00. Now the dividend is 500. To find the next divisor, we need to double our quotient obtained before. Doubling 8 gives 16; hence consider it as the next divisor.

Step 5

Now pair 16 with another number to make a new divisor that results in $\leq$ 500 when multiplied with the divisor. 

Step 6

Adding 3 to the divisor and multiplying 163 with 3 results in 489 $\leq$ 500. The remainder obtained is 11. Move the next pair of zeros down and repeat the same process mentioned above.

Step 7

Keep on repeating the same steps till the zero remainder is obtained or if the division process continues infinitely, solve to two decimal places.

Step 8

The resulting quotient of 8.306 is the square root of 69. Figure 1 given below shows the long division process in detail:

Square root of

Figure 1

Square Root by Approximation Method

The approximation method involves guessing the square root of the non-perfect square number by dividing it by the perfect square lesser or greater than that number and taking the average.

The given detailed steps must be followed to find the square root of 69 using the approximation technique.

Step 1

Consider a perfect square number of 64 less than 69.

Step 2

Now divide 69 by √64.

69 ÷ 8 = 8.625

Step 3

Now take the average of 8 and 8.625. The resulting number is approximately equivalent to the square root of 69.

(8 + 8.625) ÷ 2 = 8.31

Important points

  • The number 69 is not a perfect square.
  • The number 69 is an irrational number.
  • The number 69 can be split into its prime factorization.

Is Square Root of 69 a Perfect Square?

The number 69 is not a perfect square. A number is a perfect square if it splits into two equal parts or identical whole numbers. If a number is a perfect square, it is also rational.

A number expressed in p/q form is called a rational number. All the natural numbers are rational. A square root of a perfect square is a whole number; therefore, a perfect square is a rational number.

A number that is not a perfect square is irrational as it is a decimal number. As far as 69 is concerned, it is not a perfect square. It can be proved as below:

Factorization of 69 results in 3 x 23.

Taking the square root of the above expression gives:

= √(3 x 23)

= (3 x 23)$^{1/2}$

= 8.306

This shows that 69 is not a perfect square as it has decimal places; hence it is an irrational number.

Therefore the above discussion proves that the square root of 69 is equivalent to 8.306.

Images/mathematical drawings are created with GeoGebra.

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