Linear Sentences in One Variable
Radicals and Complex Numbers
When dividing radical expressions, use the quotient rule.
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Radical expressions are written in simplest terms when
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The index is as small as possible.
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The radicand contains no factor (other than 1) which is the nth power of an integer or polynomial.
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The radicand contains no fractions.
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No radicals appear in the denominator.
Example 1: Simplify each of the following.
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Rationalizing the denominator
An expression with a radical in its denominator should be simplified into one without a radical in its denominator. This process is called rationalizing the denominator. This is accomplished by multiplying the expression by the value 1 in an appropriate form.
Example 2: Simplify each of the following.
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What can be multiplied with
so the result will not involve a radical? The answer is
or
. Therefore,
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Conjugates
If
a and
b are unlike terms, then the
conjugate of
a + b is
a − b, and the conjugate of
a − b is
a + b. The conjugate of
is
. Conjugates are used for rationalizing the denominator when the denominator is a two-termed expression involving a square root.
Example 3: Simplify
.
To rationalize the denominator of this expression, multiply by one in the form of the conjugate over itself.
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