Square Root of 169
Sandra plans on building a square pool in her backyard. She wants the pool to have an area of 169 square feet. She is trying to calculate the side of the pool. To calculate the side of a squareshaped object, we have to find the square root of the given area. Square root of 169 is the number which when multiplied by itself gives the product as 169. In this lesson, we will calculate the square root of 169 by long division method and solve a few interesting problems.
 Square Root of 169: 13
 Square of 169: 28,561
What Is the Square Root of 169?
 The square root of a number is the number that when multiplied to itself gives the original number as the product.
 169 = a × a = a^{2 }
 Thus, a = √169 = √(13 × 13)
 13 × 13 = 169 or 13 × 13 = 169
 Thus the square root of 169 is +13 or 13.
 This shows that 169 is a perfect square.
Is the Square Root of 169 Rational or Irrational?
 A rational number is defined as a number that can be expressed in the form of a quotient or division of two integers, i.e. p/q, where q is not equal to 0.
 13 and 13 can be expressed as 13/1 and 13/1
 Both numbers can be represented in the form of a rational number.
 Hence, the square root of 169 is a rational number.
How to Find the Square Root of 169?
The square root of 169 can be calculated using different methods such as prime factorization or the long division method or the repeated subtraction method.
Square Root of 169 by Repeated Subtraction Method
Start from 169 and keep subtracting the successive odd numbers till we obtain zero. The total numbers we subtract is the square root of 169.
 169  1 = 168
 168  3 = 165
 165  5 = 160
 160  7 = 153
 153  9 = 144
 144 11 = 133
 133 13 = 120
 120 15 = 105
 105  17 = 88
 88  19 = 69
 69  21 = 48
 48  23 = 25
 25  25 = 0
Thus starting from 169, we have subtracted 13 times to obtain 0. Thus, the square root of 169 is 13.
Square Root of 169 by Long Division Method
 Write 169. Take the number in pairs from the right. 69 is chosen as a pair and one stands alone.
 Now divide 1 with a number such that number × number gives 1 or a number lesser than that.
 Obtain 1 as the quotient and 0 as the remainder.
 Bring down 69 for division.
 Multiply the quotient obtained in the previous step with 2. Now 1 × 2 = 2; 20 is written in the divisor's place.
 Find a number which can be added to 20 and this sum can be multiplied by that number such that the product is 69 or lesser than that.
 We find that 3 is the number. 20 + 3 = 23 and 23 × 3 gives 69.
 Now our quotient is 13 and the remainder is 0.
Explore Square roots using illustrations and interactive examples
Important Notes
 The square root of 169 is expressed as √169 in the radical form and as 169^{½} in exponential form.
 The square root of a number is both negative and positive for the same numerical value, i.e., the square root of 169 is +13 or 13.
 The square root of only perfect squares can be calculated easily using the prime factorization method or repeated subtraction method. 169 is a perfect square.
Challenging Questions
 Thomas is replacing the shutters on a square window of area 16.9 square inches. How high will the shutters be?
 In a class, each of the students contributed as many dollars as the total number of students. If the total collection was $16900, how many students were there in the class?
Square Root of 169 Solved Examples

Example 1: A new square playground is being built on an area of 676 square yards. The ground will be divided into four equal sections. What will be the length of each side of the section?
Solution:
The total area of the park = 676 square yards
Total sections of the park = 4
Area of each section of the park = Total area of the park ÷ Number of sections
676 ÷ 4 = 169 square yards
We know that each side of the square = √169
Thus, the length of each side of the section = 13 yards

Example 2 : Shane has a square blanket that has an area of 256 square inches. The blanket has an embroidered border of a certain width. The interior area of the blanket is 169 square inches. Can you calculate the width of the embroidery of the blanket?
Solution:
We know that length of each side of the blanket = √ 256 = 16 inches
Length of the interior of the blanket = √169 = 13 inches
Thus, width of the embroidery = 16  13 = 3 inches
FAQs on Square Root of 169
What is the square root of 169?
The square root of 169 is +13 or 13.
Why is the square root of 169 negative 13?
169 =  13 x 13. Thus, the square root of 169 can be negative 13.
Is 169 an irrational number?
No, 169 is a rational number. It can be expressed as 169/1
Between which two numbers does the square root of 169 lie?
The square root of 169 lies between 12 and 14.
What are the methods to find the square root of 169?
We can find the square root of 169 using these methods: prime factorization method, long division method, or repeated subtraction method.
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