Square Root of 169 + Solution With Free Steps

The square root of any number, in this case, the number is 169 is such that when its square root is multiplying with it again like √169x√169 equal to the number for which we are taking square root like in this case 169. 

In this article, we will analyze and find the square root of 169 using various mathematical techniques, such as the approximation method and the long division method.

What Is the Square Root Of 169?

The square root of the number 169 is 13.

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:

√169 = √(13 x 13)

√169 = √(13)$^2$

√169 = ±13

The square can be canceled with the square root as it is equivalent to 1/2; therefore, obtaining 13. Hence 13 is 169’s square root. The square root generates both positive and negative integers.

How To Calculate the Square Root of 169?

You can calculate the square root of 169 using techniques in mathematics; the Long Division method.

The symbol √ is interpreted as 169 raised to the power 1/2. So any number, when multiplied by 13 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 169 using the long division method:

Step 1

First, write the given number 169 in the division symbol, as shown in figure 1.

Step 2

Starting from the right side of the number, divide the number 169 into pairs such as 69 and 1.

Step 3

Now divide the digit 1 by a number, giving a number 1. Therefore, in this case, the remainder is zero, whereas the quotient is 1.

Step 4

After this, bring down the next pair 69. Now the dividend is 69. To find the next divisor, we need to double our quotient obtained before. Doubling 1 gives 2; hence consider it as the next divisor.

Step 5

Now pair 2 with another number to make a new divisor that results in $\leq$ 69 when multiplied with the divisor.

Step 6

Adding 3 to the divisor and multiplying 23 with 3 results in 69 $\eq$ 69. The remainder obtained is 0. 

Step 7

The resulting quotient 13 is the square root of 169. Figure 1 given below shows the long division process in detail:

figure 1

Important points

• The number 169 is a perfect square.
• The number 169 is a rational number.
• The number 169 can be split into its prime factorization.

Is Square Root of 169 a Perfect Square?

The number 169 is 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 169 is concerned, it is a perfect square. It can be proved as below:

Factorization of 169 results in 13 x 13 which can also be expressed as 13$^2$.

Taking the square root of the above expression gives:

= √(13$^2$)

= (13$^2$)$^{1/2}$

= 13

This shows that 169 is a perfect square and a rational number.

Therefore the above discussion proves that the square root of 169 is equivalent to 13.