Download 6-3 Computing with Radicals

6-3 Computing with Radicals
When adding or subtracting with radicals, you should treat the radical the same way you
would treat a variable. That is, you cannot compute it unless they are like terms. Just as
you cannot add x and y, or x2 and x3, so too, you cannot add 2 and 3 .
For instance,
Simpify
2 3 57 3 2
Because the radical are the same, i.e.
3 , we can add those like terms together.
After simplifying we get:
9 3  3 Simply adding the constants, 5 and -2, and the radicals.
Let’s try another one.
Simplify
10 2  3 2  7  6 2
13 2  7
Easy enough!
Now, what if we have radicals that are not like terms, but also not in their simplest form?
3 27  7 3  12 Here we want to factor each radical into its simplest form.
3 32  3  7 3  2 2  3
33 3  7 3  2 3
Notice that we factored out, so that we had three like terms.
9 37 32 3
0
This process works regardless of the root power, and regardless of whether we are
dealing with constants or variables. For instance,
Simplify
3
16a  43 54a
3
23  2a  43 33  2a
23 2a  4  33 2a
143 2a
Same process as above, despite being a cube root, and a mixed
number.
When multiplying radicals, we can treat them like any other binomial. If appropriate,
FOIL can be used. For instance,
Simplify

6 3

3 2

18  12  9  6
32  2  2 2  3  32  6
3 2  2 3 3 6
Or,
Simplify
2
34 5

36 5

2 9  12 15  4 15  24 25
6  12 15  4 15  120
Use FOIL. Combine like terms.
126  16 15
New term: conjugates
In the following example, the two binomials are called conjugates of each other. When
conjugates are multiplied together, assuming all numbers are rational, the resulting
answer will be rational.
Simplify
12  3 12  3 
144  12 3  12 3  32
144  9
141
Notice the middle terms cancel out. And the last term creates
a square root of a squared number. Therefore, our resulting
answer will be a rational number, i.e. devoid of any root signs.
We can use this new found knowledge to rationalize, i.e. get rid of root signs, from the
denominator of an equation.
Simplify
1 2 5 1 2 5 6  5


6 5
6 5 6 5
6  5  12 5  2 25
36  25
6  13 5  10
31
16  13 5
31
Here we simply multiply top and bottom by the conjugate
of the denominator. This turns the denominator into a
rational number, thereby simplifying the fraction.