# What is the square root of 130?

Sep 17, 2015

The actual answer is a number between 11 and 12, as $121 < 130 < 144$ so $\sqrt{{11}^{2}} < \sqrt{130} < \sqrt{{12}^{2}}$.

But it's usually bad form to evaluate the root as it'll just give us an ugly number, we'll need to put everything as approximate because you can't put the exact value of a root, etc. so it's often not really worth the trouble.

What we can do, is factor the numbers to see if there's a way to get a smaller number under the root.

While factoring we only check for primes and work from the smallest (2) to the biggest. You don't have to do it that way, but this way is the simplest as you'll cover every base and won't forget a number or so.

To factor we list the number and put a bar next to it

130 |

Then we put the smallest prime that 130 can be perfectly divided by, on th eother side of the bar, and the quotient under the number

130 | 2
65 |

And so on until we reach 1. Remembering those shortcuts to see if a number will divide or not is helpful here (i.e.: all evens are divisable by 2, all numbers that end in 5 or 0 are divisable by 5, if the sum or every digit is 3, 6 or 9 it's divisable by 3, and so on.)

In the end it comes out to

130 | 2
65 | 5
13 | 13
1 | / 130 = 2513

Since none of these numbers is a perfect square, we can't take anything out of the root. So for most cases just saying $\sqrt{130}$ is that, should suffice.

If your teacher really wants a value, you can use that range above and start estimating values, if you don't have a calculator. I.e.:

$11 < \sqrt{130} < 12$
Since 130 is closer to 121 than to 144, we can guess that it's root will be closer to 11 than to 144. We check out then with 11,5.

$11.5 \cdot 11.5 = 132.25$
$132.25 > 130$
$11 < \sqrt{130} < 11.5$

So we found a better upper range, now, since 132,25 is closer to 130 than 121, we can guess that the root will be closer to 11.5 than to 11. So we can test out with 11.4

$11.4 \cdot 11.4 = 129.96$/
$129.96 < 130$
$11.4 < \sqrt{130} < 11.5$

And so on, until we get a good enough estimate. If you have a calculator you can just put that in and find the value. Which is approximately $11.401754$