Square Root of 772 (√772)

Square Root of 772

Here we will define, analyze, simplify, and calculate the square root of 772. We start off with the definition and then answer some common questions about the square root of 772. Then, we will show you different ways of calculating the square root of 772 with and without a computer or calculator. We have a lot of information to share, so let's get started!

Square root of 772 definition
The square root of 772 in mathematical form is written with the radical sign like this √772. We call this the square root of 772 in radical form. The square root of 772 is a quantity (q) that when multiplied by itself will equal 772.

√772 = q × q = q²

Is 772 a perfect square?
772 is a perfect square if the square root of 772 equals a whole number. As we have calculated further down on this page, the square root of 772 is not a whole number.

772 is not a perfect square.

Is the square root of 772 rational or irrational?
The square root of 772 is a rational number if 772 is a perfect square. It is an irrational number if it is not a perfect square. Since 772 is not a perfect square, it is an irrational number. This means that the answer to "the square root of 772?" will have an infinite number of decimals. The decimals will not terminate and you cannot make it into an exact fraction.

√772 is an irrational number

Can the square root of 772 be simplified?
You can simplify 772 if you can make 772 inside the radical smaller. We call this process "to simplify a surd". The square root of 772 can be simplified.

√772 = 2√193

How to calculate the square root of 772 with a calculator
The easiest and most boring way to calculate the square root of 772 is to use your calculator! Simply type in 772 followed by √x to get the answer. We did that with our calculator and got the following answer with 9 decimal numbers:

√772 ≈ 27.784887979

How to calculate the square root of 772 with a computer
If you are using a computer that has Excel or Numbers, then you can enter SQRT(772) in a cell to get the square root of 772. Below is the result we got with 13 decimals. We call this the square root of 772 in decimal form.

SQRT(772) ≈ 27.7848879788996

What is the square root of 772 rounded?
The square root of 772 rounded to the nearest tenth, means that you want one digit after the decimal point. The square root of 772 rounded to the nearest hundredth, means that you want two digits after the decimal point. The square root of 772 rounded to the nearest thousandth, means that you want three digits after the decimal point.

10th: √772 ≈ 27.8

100th: √772 ≈ 27.78

1000th: √772 ≈ 27.785

What is the square root of 772 as a fraction?
Like we said above, since the square root of 772 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 772 rounded to the nearest hundredth.

√772
≈ 27.78/1
≈ 2778/100
≈ 27 39/50

What is the square root of 772 written with an exponent?
All square roots can be converted to a number (base) with a fractional exponent. The square root of 772 is no exception. Here is the rule and the answer to "the square root of 772 converted to a base with an exponent?":

√b = b^½

√772 = 772^½

How to find the square root of 772 by long division method
Here we will show you how to calculate the square root of 772 using the long division method with one decimal place accuracy. This is the lost art of how they calculated the square root of 772 by hand before modern technology was invented.

Step 1)
Set up 772 in pairs of two digits from right to left and attach one set of 00 because we want one decimal:

Step 2)
Starting with the first set: the largest perfect square less than or equal to 7 is 4, and the square root of 4 is 2. Therefore, put 2 on top and 4 at the bottom like this:

2
7	72	00
4

Step 3)
Calculate 7 minus 4 and put the difference below. Then move down the next set of numbers.

2
7	72	00
4
3	72

Step 4)
Double the number in green on top: 2 × 2 = 4. Then, use 4 and the bottom number to make this problem:

4? × ? ≤ 372

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:

47 × 7 = 329.

Now, enter 7 on top, and 329 at the bottom:

2	7
7	72	00
4
3	72
3	29

Step 5)
Calculate 372 minus 329 and put the difference below. Then move down the next set of numbers.

2	7
7	72	00
4
3	72
3	29
0	43	00

Step 6)
Double the number in green on top: 27 × 2 = 54. Then, use 54 and the bottom number to make this problem:

54? × ? ≤ 4300

The question marks are "blank" and the same "blank". With trial and error, we found the largest number "blank" can be is 7. Now, enter 7 on top:

2	7	7
7	72	00
4
3	72
3	29
0	43	00

That's it! The answer is on top. The square root of 772 with one digit decimal accuracy is 27.7. 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 772 on this page.

Notes
Remember that negative times negative equals positive. Thus, the square root of 772 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 773
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