The first step to solving a radical equation is to factor the radicand
(number under the radical) into its prime factors. Here is an example:
Now, you might already know the answer to ,
but for sake of illustration, we will start with an easy problem.
36 factors into 2 *2 *3*3. Leave this under the radical:  2*2*3*3
. The index of a square root is 2, even though it is not written. The
index is two, and this means that for every two of a factor under the
radical, we can simplify by bringing one out.
There are two 2’s under the radical: 2*2*3*3.
For that pair, since the index is 2, we can take out one 2: 2 3*3
Since there are also two 3’s, you can take one out. Now you have:
2*3
Notice that the factors you brought out are multiplied. There is nothing
left under the radical. The answer is 6.
You can also do the problem using exponents. This makes it much easier
when the radicand is a large number. Instead of counting pairs of numbers,
you divide the exponent of the factor by the index. In ,
the factored form is 22*32. Since the exponents
are 2 and the index is 2, when you divide, you get 1. This is the exponent
of the factors you bring out of the equation: 2*3 . This simplifies to
6.
What happens, though, when the radicand is not a perfect square? Here
is an example:
48
Start by factoring 48 into a product of primes. You get:
24
* 3
Since there are four 2’s under the radical, you divide the exponent
(4) by the index (2). The answer is 2. This is the exponent of the 2 after
your take it out of the radical.
223
Since there is only one three, you cannot simplify it. Simplify your
exponents, and you have:
43
Example 1
56
Start with factoring: 23
* 7
Since there are three 2’s, this can be simplified. 3/2 is
1 remainder 1. Take out one 2. Since there was a remainder of one,
leave one in the radical.
22
* 7
Multiply:
214
|
Example 2
256
Start with factoring: 25*32
Take out two 2’s and one 3. There is one 2 left inside the
radical.
22*32
Simplify:
122
|
Indexes Higher Than 2
You can do the same process when the index is higher than 2. Here is
an example:
Start by factoring the radicand. You get:
 24
*3
Since there are four 2’s, you can simplify. You divide the exponent
(4) by the index (3) and you get 1 remainder 1. The quotient is the exponent
of the 2 that you remove. The remainder is the exponent of the 2 you leave
under the radical.
2 2*3
Since there is only one two and one three left, you cannot simplify.
Multiply:
2 6
This is the simplest possible form of
Here is another example:
Start by factoring:
 38
Divide 8 by 4. You get 2. This is the exponent of the 3 that you remove.
There are no numbers left under the radical.
Example 1
256
Start by factoring:
25*32
Simplify the 2. 5 divided by 4 is 1 remainder 1
2 2*32
Multiply
2 18
|
Example 2
Start by factoring
24*3
Simplify
2 2*3
Multiply
2 6
|
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