There are three rules that must be observed in order to fully simplify any radical expression:
1) all perfect powers of the radicand must be simplified according to the product rule (explained below),
2) the radicand must have no fractions, this can be easily circumvented using the quotient rule (explained below) and
3) no denominator can contain a radical. The third checkpoint could
pose a problem.
(1) The Product Rule for Radicals states that for two radicals which have the same index, you may
multiply the two radicands and place the product under a radical with the
same index as the two original radicals. For example, . Likewise, you may “factor” a radical into two
radicals, each with the same index as the given radical. For example,
. The use of this rule allows for simplification of
radicals. For example, . When we simplify radicals using the product rule, our
goal is to factor the largest perfect power that is contained in the
radicand. If we are simplifying square roots, we are trying to find two
factors of the given radicand one of which is the largest perfect square
number contained in the radicand and the other factor containing no
perfect square number. For example, . It is important that you find the largest perfect
square factor. If you do not find the largest perfect square factor, the
radical will not be fully simplified.
(2) The Quotient Rule for
Radicals is similar to the product rule. If you have two
radicals under the same index, you may divide those radicands under a
radical of the same index as that of the two individual radicals. For
example, . Likewise, .
(3) Rationalize the
To "rationalize" the denominator multiply the denominator by so that to maintain the equality of the fraction. So now we would have
... Now the fraction is "fully simplified"
... Notice that the numerator is also a radical, so multiply
... Simplify the numerator and denominator
The views and
opinions expressed in this page are strictly those of Mary Lou Baker. This page was
The views and
opinions expressed in this page are strictly those of Mary Lou Baker.
This page was edited on 15-Nov-2007