A. List of Squares/Cubes/Etc. To Memorize
5. Higher Powers
4. Fifth Powers
3. Fourth Powers
2. Cubes
1. Squares
7.3A Simplifying Roots and Radicals
1
B. Simplifying Roots
To simplify a root, we remove the largest perfect power from inside the root and put it as the base outside. Note: If you don't remove the largest, your root may not be fully simplified! Example 1: Solution We identify the largest number in the cube table that goes into Since Since Ans
, the
comes out as a . Thus, we have
Solution We identify the largest number in the fourth power table that goes into Since Since Ans
, the
comes out as a . Thus, we have
2
is the largest number to go into
, we write
Example 2:
Simplify
.
is the largest number to go into
, we write
Simplify
.
.
.
Example 3: Solution
Simplify
We identify the largest number in the square table that goes into Since Since Ans
is the largest number to go into , the
, we write
comes out as a
. Thus, we have
C. Simplifying Radicals
1. First simplify the coefficient root. 2. To simplify the variables, use the "divide and remainder" trick: divide the power by the index; the quotient is the power that comes out and the remainder is the power that stays in. Note: For simplicity, we will assume during the next few sections that all variables represent positive real numbers so that we don't have absolute value issues. See the comments at the end.
3
.
.
D. Examples
Example 1: Simplify . Assume that all variables represent positive real numbers. Solution 1. First simplify the coefficient root: The largest fourth power that goes into Thus, we have
is
2. Now simplify the variables (divide and remainder trick): For , For , For ,
R
R
R
Ans
Thus we have
4
. .
Example 2: Simplify . Assume that all variables represent positive real numbers. Solution 1. First simplify the coefficient root: The largest cube that goes into Thus, we have is
2. Now simplify the variables (divide and remainder trick): For , For , ,
For
R
R
R
For ,
R
Example 3: Simplify . Assume that all variables represent positive real numbers. Solution 1. First simplify the coefficient root: The largest square that goes into is
Ans
5
Thus we have
.
. .
Thus, we have
2. Now simplify the variables (divide and remainder trick): For , For , For ,
R
R
R
Thus we have Ans
E. Comments
1. An alternate way to simplify roots is to write the prime factorization of the number, and then do the "divide and remainder" trick like what is used for variables. This procedure is longer, but useful if you can't figure out how to break down the root. For example, to find
:
Now use the divide and remainder trick: For , we have For , we have
6
Thus
R R
.
2. If you use the alternate method to find roots, it doesn't matter "how" you factor the number into prime powers. It is a fact that the final factorization (up to order) must be the same no matter how you do it. This fact is called the Fundamental Theorem of Arithmetic. 3. The justification for the simplification method given in this section comes from the &qu...