Here we will define, analyze, simplify, and calculate the square root of 312. We start off with the definition and then answer some common questions about the square root of 312. Then, we will show you different ways of calculating the square root of 312 with and without a computer or calculator. We have a lot of information to share, so let's get started!
Square root of 312 definition
The square root of 312 in mathematical form is written with the radical sign like this √312. We call this the square root of 312 in radical form. The square root of 312 is a quantity (q) that when multiplied by itself will equal 312.
√312 = q × q = q2
Is 312 a perfect square?
312 is a perfect square if the square root of 312 equals a whole number. As we have calculated further down on this page, the square root of 312 is not a whole number.
312 is not a perfect square.
Is the square root of 312 rational or irrational?
The square root of 312 is a rational number if 312 is a perfect square. It is an irrational number if it is not a perfect square. Since 312 is not a perfect square, it is an irrational number. This means that the answer to "the square root of 312?" will have an infinite number of decimals. The decimals will not terminate and you cannot make it into an exact fraction.
√312 is an irrational number
Can the square root of 312 be simplified?
You can simplify 312 if you can make 312 inside the radical smaller. We call this process "to simplify a surd". The square root of 312 can be simplified.
√312 = 2√78
How to calculate the square root of 312 with a calculator
The easiest and most boring way to calculate the square root of 312 is to use your calculator! Simply type in 312 followed by √x to get the answer. We did that with our calculator and got the following answer with 9 decimal numbers:
√312 ≈ 17.663521733
How to calculate the square root of 312 with a computer
If you are using a computer that has Excel or Numbers, then you can enter SQRT(312) in a cell to get the square root of 312. Below is the result we got with 13 decimals. We call this the square root of 312 in decimal form.
SQRT(312) ≈ 17.6635217326557
What is the square root of 312 rounded?
The square root of 312 rounded to the nearest tenth, means that you want one digit after the decimal point. The square root of 312 rounded to the nearest hundredth, means that you want two digits after the decimal point. The square root of 312 rounded to the nearest thousandth, means that you want three digits after the decimal point.
10th: √312 ≈ 17.7
100th: √312 ≈ 17.66
1000th: √312 ≈ 17.664
What is the square root of 312 as a fraction?
Like we said above, since the square root of 312 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 312 rounded to the nearest hundredth.
√312
≈ 17.66/1
≈ 1766/100
≈ 17 33/50
What is the square root of 312 written with an exponent?
All square roots can be converted to a number (base) with a fractional exponent. The square root of 312 is no exception. Here is the rule and the answer to "the square root of 312 converted to a base with an exponent?":
√b = b½
√312 = 312½
How to find the square root of 312 by long division method
Here we will show you how to calculate the square root of 312 using the long division method with one decimal place accuracy. This is the lost art of how they calculated the square root of 312 by hand before modern technology was invented.
Step 1)
Set up 312 in pairs of two digits from right to left and attach one set of 00 because we want one decimal:
| 3 | 12 | 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 | 12 | 00 | |
| 1 | |||
Step 3)
Calculate 3 minus 1 and put the difference below. Then move down the next set of numbers.
| 1 | |||
| 3 | 12 | 00 | |
| 1 | |||
| 2 | 12 | ||
Step 4)
Double the number in green on top: 1 × 2 = 2. Then, use 2 and the bottom number to make this problem:
2? × ? ≤ 212
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 | 12 | 00 | |
| 1 | |||
| 2 | 12 | ||
| 1 | 89 | ||
Step 5)
Calculate 212 minus 189 and put the difference below. Then move down the next set of numbers.
| 1 | 7 | ||
| 3 | 12 | 00 | |
| 1 | |||
| 2 | 12 | ||
| 1 | 89 | ||
| 0 | 23 | 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? × ? ≤ 2300
The question marks are "blank" and the same "blank". With trial and error, we found the largest number "blank" can be is 6. Now, enter 6 on top:
| 1 | 7 | 6 | |
| 3 | 12 | 00 | |
| 1 | |||
| 2 | 12 | ||
| 1 | 89 | ||
| 0 | 23 | 00 | |
That's it! The answer is on top. The square root of 312 with one digit decimal accuracy is 17.6. 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 312 on this page.
Notes
Remember that negative times negative equals positive. Thus, the square root of 312 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 313
Here is the next number on our list that we have equally detailed square root information about.
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