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

$\sqrt{169}=13$

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:

$x={n}^{2}$

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 $169$.

To solve $\sqrt{169}$, 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.

Prime Number ($n$) | $169\xf7n$ | 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:

$\sqrt{169}=\sqrt{13*13}=\sqrt{{13}^{2}}$

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 $2$:

$\sqrt{{13}^{\frac{2}{2}}}=\sqrt{{13}^{1}}$

Move ${13}^{1}$ outside the radical sign and leave $1$ under the radical sign (since $\sqrt{169}$ cannot be equal to *nothing*)

$\sqrt{{13}^{1}}={13}^{1}\sqrt{1}$

${13}^{1}\sqrt{1}$ 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:

- ${13}^{1}=13$
- $\sqrt{1}=1$

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.

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

You probably instantly recall ${12}^{2}$ is $144$. Is 169 more or less than 144? It is more, so you need a bigger number.

Let's try $15$. We calculate ${15}^{2}=225$, 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:

${14}^{2}=196$ – getting closer!

${13}^{2}=169$

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 ($169$):

$13*13=169$

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.

$169=\stackrel{-}{01}\stackrel{-}{69}$

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:

$1\phantom{\rule{0ex}{0ex}}\stackrel{-}{01}\stackrel{-}{69}\phantom{\rule{0ex}{0ex}}1\phantom{\rule{0ex}{0ex}}\stackrel{-}{0}$

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

$1\phantom{\rule{0ex}{0ex}}\stackrel{-}{01}\stackrel{-}{69}\phantom{\rule{0ex}{0ex}}1\phantom{\rule{0ex}{0ex}}\stackrel{-}{0}\stackrel{-}{69}$

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

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

$2\underset{\_}{?}\times \underset{\_}{?}\le 69$

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

$2\underset{\_}{3}\times \underset{\_}{3}=69$

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 $169$.

$\mathbf{}\mathbf{1}\mathbf{}\mathbf{}\mathbf{}\mathbf{3}\phantom{\rule{0ex}{0ex}}\stackrel{-}{01}\stackrel{-}{69}\phantom{\rule{0ex}{0ex}}1\phantom{\rule{0ex}{0ex}}\stackrel{-}{0}\stackrel{-}{69}\phantom{\rule{0ex}{0ex}}069$

The answer is $13$.

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.

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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