What Is The Square Root Of 169?

The square root of 169 is 13 – this is technically the principal square root of 169.

169 = 13

How To Find The Square Root Of 169

When we approach the problem like this, we are finding the square root (sqrt) of the original number. The square root of x is a number, n, that satisfies this equation:

x = n2

Every positive number has two square roots, a positive and a negative.

The positive square root is called the principal square root, and it is generally understood to be the one we are interested in finding.

In this case, we want the positive square root of 169.

Finding Square Root Of 169 By Factoring

To solve 169, we can factor the number under the radical sign. Prime factorization of the first several prime numbers will help us see which ones might be factors of the target number.

Finding Factors Of 169
Prime Number (n) 169 ÷ n Does It Factor?
2 84.5 No
3 56.33 No
5 33.8 No
7 24.14286 No
11 15.36 No
13 13 Yes!
17 9.941 No
Only one prime number is a factor, 13, and it times itself gives our target number, 169.

Rewrite the original problem, substituting our found factor for the original number:

169 = 13 * 13 = 132

We cannot leave the exponent under the radical. To remove exponents from under radical signs, divide even exponents by 2, then move the base (in this case, 13) and its resulting exponent outside the radical sign.

Divide the exponent 2 by 2:

1322 = 131

Move 131 outside the radical sign and leave 1 under the radical sign (since 169 cannot be equal to nothing)

131 = 1311

1311 is the square root of 169 in radical form – this is the simplest radical form

The new result looks complicated, but if you simplify the parts, you find it is not:

  • 131 = 13
  • 1 = 1

This yields 13 * 1, which is 13.

The principal square root of 169 is 13.

The square root of 169 is a rational number because it is a perfect square -- the answer has no decimals.

How Do You Find The Square Root Of 169 By Estimating?

Another way to find the square root of 169 is to estimate using known squares. 

You probably instantly recall 122 is 144. Is 169 more or less than 144? It is more, so you need a bigger number.

Let's try 15. We calculate 152 = 225, which is too big.

We have now learned the square root of 169 is somewhere between 12 and 15. Try the two remaining whole numbers:

142 = 196 – getting closer!

132 = 169

The square root of 169 is 13.

While -13 is also the square root of 169 in practice, we are always searching for the principal square root. Unless asked to find “all” the square roots of 169, we keep with our answer of 13.

You can verify that you have the correct square root of a number by multiply the number (13) times itself (13) to see if it equals the target number (169):

13 * 13 = 169

Square Root Of 169 By Division Method

Long division is another way to find a square root of 169 with a calculator.

To start, work from right to left and split the real number 169 into two pairs of two-digit numbers.

169 = 01- 69-

Now, you work on each pair separately. What is the largest perfect square less than or equal to 1? The answer is 1. And the square root of 1 is also 1.

So, we put one on top and on the bottom like this:

 1 01- 69- 1  0-

Now, subtract 1 from one and bring the remaining 69 down with your answer:

 1 01- 69- 1  0- 69-

The next step is to double the number at the very top, which is 1. So, 1 × 2 = 2.

Then, you use the number 2 and the remaining number at the bottom (69) to create this math problem:

2?_ × ?_   69

Try and find which number fits to complete the equation. The largest number that works is 3. This means we have:

23_ × 3_  = 69

Now, you can add 3 to the top of your long division problem and 69 at the bottom; 69 minus 69 is 0, so you are done. The top is your square root of 169.

 1   301- 69- 1   0- 69- 0 69

The answer is 13.

What you learned:

After working your way through this lesson and video, you have learned:

  • That the square root of 169 is 13.
  • Every positive number has a positive and negative square root – the positive square root is called the principal root.
  • How to find the square root of 169 by factoring and estimating
  • How to check your work
Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

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