Back To CourseELM: CSU Math Study Guide
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No matter where you live, the language of mathematics is the same. You might not be able to order dinner in a restaurant in South America, but by using the unique language that is math, you can work problems with anyone from any country. This is why it is very important that mathematical equations get treated the same no matter who is writing them.
The term 'radical' is just another way to say 'square root.' When writing square roots in correct mathematical language, it is important that every radical is written in its simplest form. This applies to both the numbers and variables that are under the square root symbol.
Below you see some examples of radicals that contain both numbers and variables. Some of them are simplified and others aren't. Can you tell which examples need to be simplified further?
Of these four examples, numbers 1, 3 and 4 can all be simplified further.
Here are the steps to simplifying a square root with a number:
1. First, factor the number completely. An easy way to factor a number is by using a factor tree. A factor tree can be created by writing down the number you want to factor and drawing two lines coming down from that number. Then, write two factors of that number under the lines. Continue on in this manner until only prime numbers remain. A prime number is one that cannot be reduced any smaller. The purpose of the factor tree is to determine which numbers can be removed from under the square root symbol.
2. Match up pairs of the same number. Any numbers with a partner are perfect squares and you can take the square root of those numbers.
3. Numbers without a partner remain under the square root symbol. These numbers cannot be simplified further.
Let's return to the examples from earlier and look only at the number portion of some of them.
The first example is √81x^4.
Looking at just the number portion, factor 81. 9*9 is 81, and then 3*3 is 9; therefore the factorization of 81 is 3*3*3*3, which is two groups of 3. Each group means a 3 will be removed from underneath the radical, which means that the square root of 81 is 3*3, or 9.
The √13xy is the second example.
Since 13 is a prime number, it cannot be factored and therefore is as simple as it can be and no changes can be made.
Before we can talk about finding the square root of a variable, we should probably review what exactly the square root is. The square root operation is the opposite of squared. This works for both numbers and variables. So, x*x = x^2, and by performing the opposite operation, √x^2 = x.
Simplifying square roots of variables works about the same way as it does with numbers. Just like you can factor numbers, variables with exponents can also be factored.
For example, x^4 is the same as x*x*x*x.
Then, you can proceed the same way as simplifying the square root of numbers.
1. First, match up pairs of the same variable. Using our example from before, grouping pairs of xs gives us (x*x)*(x*x) - two groups of xs.
2. Any letters with a partner are perfect squares, and you can take the square root of them. In this case, there are two pairs of xs, so there will be two xs removed from under the square root symbol.
3. Variables without a partner remain under the square root. They cannot be simplified further.
Going back again to our examples from earlier, let's look this time at the variable portion.
The first example is x^4.
If we factor x^4, we get x*x*x*x.
Break that into pairs to get (x*x)*(x*x). One x from each pair is taken out from under the square root symbol. This leaves us with an x^2 outside of the square root.
The final simplification of √81x^4 is 9x^2.
The second example is √13xy, and as you can see, there is only one x and one y under the square root symbol. This means that it cannot be simplified any further; it is as simple as it can get.
Let's try the last two examples to put it all together. Example number 3 is √(13(x^6')y^2).
As before, the first step is to look at the number - in this case, 13. Since 13 is a prime number, it cannot be factored.
Next, we look at the variables. x^6 can be factored to (x*x)*(x*x)*(x*x). Because there are three groups of xs, three xs will come out from underneath the radical symbol.
y^2 is y*y, which is one group of ys. Therefore, one y can be removed from under the square root symbol.
So, the simplification of this problem is (x^3)y√13
Let's try the last one: √((8x^2)(y^4)z).
Starting with the number, factor 8 to its smallest parts, which is 2*2*2. One pair of twos means that a 2 will come out of the square root symbol. The lone two will stay under the square root symbol.
Next, we move on to the variables. x^2 is x*x, which is a pair of xs, meaning that one x will come out of the square root.
y^4 is y*y*y*y. The two pairs of ys mean that 2 ys will come out of the square root.
Because the z is alone, it will stay underneath the square root symbol, which means that the simplification of √((8x^2)(y^4)z) is 2xy^2√(2z).
In the international world of mathematics, it is important that there is a universal language so that problems can be understood all over the world. Part of this language means that radicals, or square roots, are simplified in a certain way. To simplify a square root, first factor the numbers or variables, then pair up the like terms. Finally, remove one of each pair from under the square root symbol. Any numbers or variables without a partner remain under the square root symbol.
You should be able to simplify radicals using ordered steps after watching this video lesson.
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Back To CourseELM: CSU Math Study Guide
16 chapters | 140 lessons