What is the square root of 169?

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

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169 = 13

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How to discover the square root of 169

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


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

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

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

Finding square source of 169 by factoring

To resolve 169, we deserve to factor the number under the radical sign. Element factorization that the first several element numbers will assist us see which ones could be components of the target number.

Finding factors Of 169Prime Number (n)169 ÷ nDoes it Factor?
284.5No
356.33No
533.8No
724.14286No
1115.36No
1313Yes!
179.941No

Only one element number is a factor, 13, and it times itself provides our target number, 169.

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Rewrite the initial problem, substituting our uncovered factor because that the original number:

169 = 13 * 13 = 132

We cannot leave the exponent under the radical. To remove exponents native 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 through 2:

1322 = 131

Move 131 external the radical sign and leave 1 under the radical sign (since 169 cannot be same to nothing)131 = 1311


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

The brand-new result watch complicated, but if you leveling the parts, you discover it is not:

131 = 131 = 1

This yields 13 * 1, i beg your pardon is 13.

The primary square source 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 discover The Square source Of 169 through Estimating?

Another way to find the square source of 169 is to calculation using recognized squares.

You more than likely instantly recall 122 is 144. Is 169 more or much less than 144? it is more, so you require a enlarge number.

Let"s try 15. Us calculate 152 = 225, i beg your pardon is too big.

We have now learned the square source of 169 is somewhere between 12 and 15. Shot the 2 remaining whole numbers:

142 = 196 – getting closer!

132 = 169

The square source of 169 is 13.

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While -13 is additionally the square source of 169 in practice, us are constantly searching because that the major square root. Unless asked to discover “all” the square root of 169, we store with ours answer that 13.

You can verify the you have actually the correct square root of a number by multiply the number (13) time itself (13) to view if it amounts to the target number (169):

13 * 13 = 169

Square root of 169 by department method

Long department is another means to uncover a square source of 169 v a calculator.

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

169 = 01- 69-

Now, you occupational on each pair separately. What is the biggest perfect square less than or equal to 1? The prize is 1. And also the square source of 1 is likewise 1.

So, we placed one on top and also on the bottom like this: 1 01- 69- 1  0-

Now, subtract 1 native one and also bring the remaining 69 down through your answer: 1 01- 69- 1  0- 69-

The next step is to dual the number in ~ the really top, i beg your pardon is 1. So, 1 × 2 = 2.

Then, you usage the number 2 and the staying number in ~ the bottom (69) to create this mathematics problem:

2?_ × ?_  ≤ 69

Try and find which number fits to finish the equation. The biggest number that functions is 3. This way we have:23_ × 3_  = 69

Now, girlfriend can include 3 to the height of her long department problem and 69 in ~ the bottom; 69 minus 69 is 0, for this reason you room done. The height is your square root of 169.

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 1   301- 69- 1   0- 69- 0 69

The answer is 13.


What girlfriend learned:

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

That the square root of 169 is 13.Every hopeful number has actually a confident and an unfavorable square source – the confident square root is referred to as the primary root.How to uncover the square root of 169 by factoring and estimatingHow to examine your work