# Square Root Of 169 — How To Find

## What is the square root of 169?

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

## 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:

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 169169.

### Finding the square root of 169 by factoring

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 ÷ 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 |

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

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**:

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

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.

### 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 ${12}^{2}$ 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 that the square root of **169** is somewhere between **12** and **15**. Try the two remaining whole numbers.

Getting closer!

The square root of **169** is **13**.

You can verify that you have the correct square root of a number by multiplying the number times itself to see if it equals the target number:

### Square root of 169 by division method

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

Work from right to left and split the real number **169** into two pairs of two-digit numbers.

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 the bottom like this:

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

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:

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

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**.

The answer is **13**.