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How to Simplify Square Roots (Radicals)

 
A common task in algebra is to simplify square roots, or what is referred to in later mathematics as radicals. This article will use the sqrt (x) notation to mean "the square root of a number x". Sometimes quite easy task to simplify, but sometimes requires the help of a special formula, together with your knowledge of perfect squares and factors. For example, this would be the case for a radical, such as sqrt (80).

It is therefore very important, because if a radical is not simplified, it is typically considered to be wrong, and you will either not receive or partial credit for your answer to an exam. This article shows you the simple steps to perform this task.

This article assumes that you are familiar with the basic concepts of square and "square rooting." See the Resource section for more information on these topics.

1. It's easy to simplify a radical, which is a perfect square, such as sqrt (81). We can either use a calculator, or we can use our knowledge of perfect squares to get a response at 9, page 9 equal 9. . We have to remember that-9 is also a solution to the problem, although it would be discarded in connection with a geometry problem involving lengths, or if we only were asked to give the principal square root.

2. Simplifying a non-perfect square radical such as sqrt (20) involves a bit more work. We could use a calculator to get a long decimal approximation of the answer, but it is not what is meant by simplifying the radicals. What we are being asked to do, in essence, is to break the radical one another in such a way that we are left with the product of an integer times the square root of a prime number.

3. To do this, it is important to know the specific property of radicals is shown above. In the simplest form is the equation tells us that we can split the radical of a product in the product of the radicals. If you want to apply the formula at sqrt (20) example above, we would break 20 of factors of 4 and 5. . We have then sqrt (4 x 5), which can be divided into sqrt (4) times sqrt (5). SQRT (4) is 2, so we know our final simplified answer is 2 times sqrt (5). It is the response that would be expected on an exam. Note how we cannot break down sqrt (5), since 5 is a prime number, whose only factors are 1 and itself.

4. Sometimes students asking if they could have broken 20 in other factors, such as 2 and 10. the answer is that we could, but we would be sqrt (2 times 10), which would break into sqrt (2) times sqrt (10). Since none of them is a perfect square, we will not end up with an integer component of our response, which is what we need.

5. Let's get back to the example of sqrt (80) in the introduction. 80 can be broken up into many factor pairs, such as 2 and 40, 4 and 20, 8 and 10, etc. What we have to look for is the largest perfect square factor 80, and use it. 4 is a perfect square factor on 80, but there is a bigger:. 16. This means that we must use 16 and 5, as our factor pair. We have now sqrt (16 times 5) = sqrt (16) times sqrt (5) = 4 times sqrt (5), as is our answer.

6. In the above example, if we had used the 4 and 20, as our factor pair, we would have lots of extra work to do. We would be sqrt (4) times sqrt (20). It will be 2 times sqrt (20), but then we would have to break down sqrt (20), as we did before. By using the largest perfect square factor 16, we got our answer in fewer steps.

7. One last example:. SQRT (200). There are many factors, several of which are perfect squares. We want the largest perfect square factor is 100. It gives us the sqrt (100) times sqrt (2), which represent 10 times sqrt (2).

8. Please note that we have no way of reducing the square root of a number, which is either the primary or the product of two primes. For example, we may not simplify the sqrt (13). It is a prime number with someone perfects square factors. We are just going to have to leave our response is.

Another example would be sqrt (6). 6 is not prime. We could break it up in sqrt (2) times sqrt (3), but none of them is a perfect square, so it will not simplify. We just wanted to let our response as sqrt (6). It does not have any perfect square factor.

A last example is sqrt (77). 77 is not prime, since it has other than 1 and itself, but the other factors are both primes. Since it has no perfect square factors, we just have to leave the answer alone, and it is right to do so.

9. Algebra students must ensure that they are comfortable with this process. It comes up quite often in mathematics, and there is no reason to make a problem perfectly, but then lose partial or full credit just because you do not simplify your square root answers.
 
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