Here we will define, analyze, simplify, and calculate the square root of 414. We start off with the definition and then answer some common questions about the square root of 414. Then, we will show you different ways of calculating the square root of 414 with and without a computer or calculator. We have a lot of information to share, so let's get started!
Square root of 414 definition
The square root of 414 in mathematical form is written with the radical sign like this √414. We call this the square root of 414 in radical form. The square root of 414 is a quantity (q) that when multiplied by itself will equal 414.
√414 = q × q = q2
Is 414 a perfect square?
414 is a perfect square if the square root of 414 equals a whole number. As we have calculated further down on this page, the square root of 414 is not a whole number.
414 is not a perfect square.
Is the square root of 414 rational or irrational?
The square root of 414 is a rational number if 414 is a perfect square. It is an irrational number if it is not a perfect square. Since 414 is not a perfect square, it is an irrational number. This means that the answer to "the square root of 414?" will have an infinite number of decimals. The decimals will not terminate and you cannot make it into an exact fraction.
√414 is an irrational number
Can the square root of 414 be simplified?
You can simplify 414 if you can make 414 inside the radical smaller. We call this process "to simplify a surd". The square root of 414 can be simplified.
√414 = 3√46
How to calculate the square root of 414 with a calculator
The easiest and most boring way to calculate the square root of 414 is to use your calculator! Simply type in 414 followed by √x to get the answer. We did that with our calculator and got the following answer with 9 decimal numbers:
√414 ≈ 20.346989949
How to calculate the square root of 414 with a computer
If you are using a computer that has Excel or Numbers, then you can enter SQRT(414) in a cell to get the square root of 414. Below is the result we got with 13 decimals. We call this the square root of 414 in decimal form.
SQRT(414) ≈ 20.3469899493758
What is the square root of 414 rounded?
The square root of 414 rounded to the nearest tenth, means that you want one digit after the decimal point. The square root of 414 rounded to the nearest hundredth, means that you want two digits after the decimal point. The square root of 414 rounded to the nearest thousandth, means that you want three digits after the decimal point.
10th: √414 ≈ 20.3
100th: √414 ≈ 20.35
1000th: √414 ≈ 20.347
What is the square root of 414 as a fraction?
Like we said above, since the square root of 414 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 414 rounded to the nearest hundredth.
√414
≈ 20.35/1
≈ 2035/100
≈ 20 7/20
What is the square root of 414 written with an exponent?
All square roots can be converted to a number (base) with a fractional exponent. The square root of 414 is no exception. Here is the rule and the answer to "the square root of 414 converted to a base with an exponent?":
√b = b½
√414 = 414½
How to find the square root of 414 by long division method
Here we will show you how to calculate the square root of 414 using the long division method with one decimal place accuracy. This is the lost art of how they calculated the square root of 414 by hand before modern technology was invented.
Step 1)
Set up 414 in pairs of two digits from right to left and attach one set of 00 because we want one decimal:
| 4 | 14 | 00 |
Step 2)
Starting with the first set: the largest perfect square less than or equal to 4 is 4, and the square root of 4 is 2. Therefore, put 2 on top and 4 at the bottom like this:
| 2 | |||
| 4 | 14 | 00 | |
| 4 | |||
Step 3)
Calculate 4 minus 4 and put the difference below. Then move down the next set of numbers.
| 2 | |||
| 4 | 14 | 00 | |
| 4 | |||
| 0 | 14 | ||
Step 4)
Double the number in green on top: 2 × 2 = 4. Then, use 4 and the bottom number to make this problem:
4? × ? ≤ 14
The question marks are "blank" and the same "blank". With trial and error, we found the largest number "blank" can be is 0. Replace the question marks in the problem with 0 to get:
40 × 0 = 0.
Now, enter 0 on top, and 0 at the bottom:
| 2 | 0 | ||
| 4 | 14 | 00 | |
| 4 | |||
| 0 | 14 | ||
| 0 | 00 | ||
Step 5)
Calculate 14 minus 0 and put the difference below. Then move down the next set of numbers.
| 2 | 0 | ||
| 4 | 14 | 00 | |
| 4 | |||
| 0 | 14 | ||
| 0 | 00 | ||
| 0 | 14 | 00 | |
Step 6)
Double the number in green on top: 20 × 2 = 40. Then, use 40 and the bottom number to make this problem:
40? × ? ≤ 1400
The question marks are "blank" and the same "blank". With trial and error, we found the largest number "blank" can be is 3. Now, enter 3 on top:
| 2 | 0 | 3 | |
| 4 | 14 | 00 | |
| 4 | |||
| 0 | 14 | ||
| 0 | 00 | ||
| 0 | 14 | 00 | |
That's it! The answer is on top. The square root of 414 with one digit decimal accuracy is 20.3. 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 414 on this page.
Notes
Remember that negative times negative equals positive. Thus, the square root of 414 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 415
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