Here we will define, analyze, simplify, and calculate the square root of 209. We start off with the definition and then answer some common questions about the square root of 209. Then, we will show you different ways of calculating the square root of 209 with and without a computer or calculator. We have a lot of information to share, so let's get started!
Square root of 209 definition
The square root of 209 in mathematical form is written with the radical sign like this √209. We call this the square root of 209 in radical form. The square root of 209 is a quantity (q) that when multiplied by itself will equal 209.
√209 = q × q = q2
Is 209 a perfect square?
209 is a perfect square if the square root of 209 equals a whole number. As we have calculated further down on this page, the square root of 209 is not a whole number.
209 is not a perfect square.
Is the square root of 209 rational or irrational?
The square root of 209 is a rational number if 209 is a perfect square. It is an irrational number if it is not a perfect square. Since 209 is not a perfect square, it is an irrational number. This means that the answer to "the square root of 209?" will have an infinite number of decimals. The decimals will not terminate and you cannot make it into an exact fraction.
√209 is an irrational number
Can the square root of 209 be simplified?
You can simplify 209 if you can make 209 inside the radical smaller. We call this process "to simplify a surd". The square root of 209 cannot be simplified.
√209 is already in its simplest radical form.
How to calculate the square root of 209 with a calculator
The easiest and most boring way to calculate the square root of 209 is to use your calculator! Simply type in 209 followed by √x to get the answer. We did that with our calculator and got the following answer with 9 decimal numbers:
√209 ≈ 14.456832295
How to calculate the square root of 209 with a computer
If you are using a computer that has Excel or Numbers, then you can enter SQRT(209) in a cell to get the square root of 209. Below is the result we got with 13 decimals. We call this the square root of 209 in decimal form.
SQRT(209) ≈ 14.4568322948010
What is the square root of 209 rounded?
The square root of 209 rounded to the nearest tenth, means that you want one digit after the decimal point. The square root of 209 rounded to the nearest hundredth, means that you want two digits after the decimal point. The square root of 209 rounded to the nearest thousandth, means that you want three digits after the decimal point.
10th: √209 ≈ 14.5
100th: √209 ≈ 14.46
1000th: √209 ≈ 14.457
What is the square root of 209 as a fraction?
Like we said above, since the square root of 209 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 209 rounded to the nearest hundredth.
√209
≈ 14.46/1
≈ 1446/100
≈ 14 23/50
What is the square root of 209 written with an exponent?
All square roots can be converted to a number (base) with a fractional exponent. The square root of 209 is no exception. Here is the rule and the answer to "the square root of 209 converted to a base with an exponent?":
√b = b½
√209 = 209½
How to find the square root of 209 by long division method
Here we will show you how to calculate the square root of 209 using the long division method with one decimal place accuracy. This is the lost art of how they calculated the square root of 209 by hand before modern technology was invented.
Step 1)
Set up 209 in pairs of two digits from right to left and attach one set of 00 because we want one decimal:
| 2 | 09 | 00 |
Step 2)
Starting with the first set: the largest perfect square less than or equal to 2 is 1, and the square root of 1 is 1. Therefore, put 1 on top and 1 at the bottom like this:
| 1 | |||
| 2 | 09 | 00 | |
| 1 | |||
Step 3)
Calculate 2 minus 1 and put the difference below. Then move down the next set of numbers.
| 1 | |||
| 2 | 09 | 00 | |
| 1 | |||
| 1 | 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? × ? ≤ 109
The question marks are "blank" and the same "blank". With trial and error, we found the largest number "blank" can be is 4. Replace the question marks in the problem with 4 to get:
24 × 4 = 96.
Now, enter 4 on top, and 96 at the bottom:
| 1 | 4 | ||
| 2 | 09 | 00 | |
| 1 | |||
| 1 | 09 | ||
| 0 | 96 | ||
Step 5)
Calculate 109 minus 96 and put the difference below. Then move down the next set of numbers.
| 1 | 4 | ||
| 2 | 09 | 00 | |
| 1 | |||
| 1 | 09 | ||
| 0 | 96 | ||
| 0 | 13 | 00 | |
Step 6)
Double the number in green on top: 14 × 2 = 28. Then, use 28 and the bottom number to make this problem:
28? × ? ≤ 1300
The question marks are "blank" and the same "blank". With trial and error, we found the largest number "blank" can be is 4. Now, enter 4 on top:
| 1 | 4 | 4 | |
| 2 | 09 | 00 | |
| 1 | |||
| 1 | 09 | ||
| 0 | 96 | ||
| 0 | 13 | 00 | |
That's it! The answer is on top. The square root of 209 with one digit decimal accuracy is 14.4. Did you notice that the last two steps repeat the previous two steps. You can add decimals by simply adding more sets of 00 and repeating the last two steps over and over.
Square Root of a Number
Please enter another number in the box below to get the square root of the number and other detailed information like you got for 209 on this page.
Notes
Remember that negative times negative equals positive. Thus, the square root of 209 does not only have the positive answer that we have explained above, but also the negative counterpart.
We often refer to perfect square roots on this page. You may want to use the list of perfect squares for reference.
Square Root of 210
Here is the next number on our list that we have equally detailed square root information about.
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