The square root of is – this is technically the principal square root of .
When we approach the problem like this, we are finding the square root (sqrt) of the original number. The square root of is a number, , that satisfies this equation:
Every positive number has two square roots, a positive and a negative.
The positive square root is called the principal square root, and it is generally understood to be the one we are interested in finding.
In this case, we want the positive square root of .
To solve , we can factor the number under the radical sign. Prime factorization of the first several prime numbers will help us see which ones might be factors of the target number.
|Prime Number ()||Does It Factor?|
Rewrite the original problem, substituting our found factor for the original number:
We cannot leave the exponent under the radical. To remove exponents from under radical signs, divide even exponents by , then move the base (in this case, ) and its resulting exponent outside the radical sign.
Divide the exponent by :
Move outside the radical sign and leave under the radical sign (since cannot be equal to nothing)
The new result looks complicated, but if you simplify the parts, you find it is not:
This yields , which is .
The principal square root of is .
The square root of is a rational number because it is a perfect square -- the answer has no decimals.
Another way to find the square root of is to estimate using known squares.
You probably instantly recall is . Is 169 more or less than 144? It is more, so you need a bigger number.
Let's try . We calculate , which is too big.
We have now learned the square root of is somewhere between and . Try the two remaining whole numbers:
– getting closer!
The square root of is .
While is also the square root of in practice, we are always searching for the principal square root. Unless asked to find “all” the square roots of , we keep with our answer of .
You can verify that you have the correct square root of a number by multiply the number () times itself () to see if it equals the target number ():
Long division is another way to find a square root of with a calculator.
To start, work from right to left and split the real number into two pairs of two-digit numbers.
Now, you work on each pair separately. What is the largest perfect square less than or equal to ? The answer is . And the square root of is also .
So, we put one on top and on the bottom like this:
Now, subtract from one and bring the remaining down with your answer:
The next step is to double the number at the very top, which is . So, .
Then, you use the number and the remaining number at the bottom () to create this math problem:
Try and find which number fits to complete the equation. The largest number that works is . This means we have:
Now, you can add to the top of your long division problem and at the bottom; minus is , so you are done. The top is your square root of .
The answer is .
After working your way through this lesson and video, you have learned:
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