# Square Root of 309

In this article we're going to calculate the square root of 309 and explore what the square root is and answer some of the common questions you might. We'll also look at the different methods for calculating the square root of 309 (both with and without a computer/calculator).

## Square Root of 309 Definition

In mathematical form we can show the square root of 309 using the radical sign, like this: √309. This is usually referred to as the square root of 309 in radical form.

So what is the square root? In this case, the square root of 309 is the quantity (which we will call q) that when multiplied by itself, will equal 309.

√309 = q × q = q2

## Is 309 a Perfect Square?

In math, we refer to 309 being a perfect square if the square root of 309 is a whole number.

In this case, as we will see in the calculations below, we can see that 309 is not a perfect square.

To find out more about perfect squares, you can read about them and look at a list of 1000 of them in our What is a Perfect Square? article.

## Is The Square Root of 309 Rational or Irrational?

A common question is to ask whether the square root of 309 is rational or irrational. Rational numbers can be written as a fraction and irrational numbers cannot.

A quick way to check this is to see if 309 is a perfect square. If it is, then it is a rational number. If it's not a perfect square then it's an irrational number.

We already know if 309 is a perfect square so we also can see that √309 is an irrational number.

## Can the Square Root of 309 Be Simplified?

309 can be simplified only if you can make 309 inside the radical symbol smaller. This is a process that is called simplifying the surd. In this example square root of 309 cannot be simplified.

√309 is already in its simplest radical form.

## How to Calculate The Square Root of 309 with a Calculator

If you have a calculator then the simplest way to calculate the square root of 309 is to use that calculator. On most calculators you can do this by typing in 309 and then pressing the √x key. You should get the following result:

√309 ≈ 17.5784

## How to Calculate the Square Root of 309 with a Computer

On a computer you can also calculate the square root of 309 using Excel, Numbers, or Google Sheets and the SQRT function, like so:

SQRT(309) ≈ 17.578395831247

## What is the Square Root of 309 Rounded?

Sometimes you might need to round the square root of 309 down to a certain number of decimal places. Here are the solutions to that, if needed.

10th: √309 ≈ 17.6

100th: √309 ≈ 17.58

1000th: √309 ≈ 17.578

## What is the Square Root of 309 as a Fraction?

We covered earlier in this article that only a rational number can be written as a fraction, and irrational numbers cannot.

Like we said above, since the square root of 309 is an irrational number, we cannot make it into an exact fraction. However, we can make it into an approximate fraction using the square root of 309 rounded to the nearest hundredth.

√309

≈ 17.6/1

≈ 1758/100

≈ 17 29/50

## What is the Square Root of 309 Written with an Exponent?

All square root calculations can be converted to a number (called the base) with a fractional exponent. Let's see how to do that with the square root of 309:

√b = b½

√309 = 309½

## How to Find the Square Root of 309 Using Long Division

Finally, we can use the long division method to calculate the square root of 309. This is very useful for long division test problems and was how mathematicians would calculate the square root of a number before calculators and computers were invented.

### Step 1

Set up 309 in pairs of two digits from right to left and attach one set of 00 because we want one decimal:

3
09
00

### Step 2

Starting with the first set: the largest perfect square less than or equal to 3 is 1, and the square root of 1 is 1 . Therefore, put 1 on top and 1 at the bottom like this:

 1 3 09 00 1

### Step 3

Calculate 3 minus 1 and put the difference below. Then move down the next set of numbers.

 1 3 09 00 1 2 09

### Step 4

Double the number in green on top: 1 × 2 = 2. Then, use 2 and the bottom number to make this problem:

2? × ? ≤ 209

The question marks are "blank" and the same "blank". With trial and error, we found the largest number "blank" can be is 7. Replace the question marks in the problem with 7 to get:

27 × 7 = 189

Now, enter 7 on top, and 189 at the bottom:

 1 7 3 09 00 1 2 09 1 89

### Step 5

Calculate 209 minus 189 and put the difference below. Then move down the next set of numbers.

 1 7 3 09 00 1 2 09 1 89 0 20 00

### Step 6

Double the number in green on top: 17 × 2 = 34. Then, use 34 and the bottom number to make this problem:

34? × ? ≤ 2000

The question marks are "blank" and the same "blank". With trial and error, we found the largest number "blank" can be is 5.

Now, enter 5 on top:

 1 7 5 3 09 00 1 2 09 1 89 0 20 00

That's it! The answer shown at the top in green. The square root of 309 with one digit decimal accuracy is 17.6. Notice that the last two steps actually repeat the previous two. To add decimal places to your answe you can simply add more sets of 00 and repeat the last two steps.