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Arithmetic Operations of Radicals
Motivation:
We still start from some obvious examples. = 2 3 = 6 And let’s check the following: = =6 So, = .
Do we always have ? We might consider in this way: we always have pnqn = (pq)n from commutative law of multiplication if n is an integer. If that can be applied to the case for rational number, then we will have
p1/nq1/n = (pq)1/n , n is an integer .
If it does not hold for rational numbers, the result for integers would not hold either because
(p1/n)n (q1/n)n = pq ((pq)1/n)n = pq
So, in general, we always have
Multiplication of Radicals
From the conclusion above, the multiplication of radicals only requires that all of the radicals are with the same denominator when they are represented in fraction exponent form.
Example 1: = ? = = =
Example 2: = =
Example 3: ( )2 = =
Example 4: = = = =
Example 5: ( )2 = 50
Addition of Radicals
The addition of radicals is just adding up the coefficients of the radicals of the same types.
Example 1: + =
However, most of the time, the terms of same type of radicals does not show up so straightforward. Let’s take a look at the second example.
Example 2: +
At the first glance, it seems that we can not do anything about it. But = = = = So. + = + =
Example 3: =
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