Download Introduction To Radicals With Higher Indices

Introduction To Radicals With Higher Indices
Square roots are not the only radicals. There are cube roots, fourth
roots, fifth roots, etc. This section will offer a brief introduction to these
higher roots.
The small number outside the radical sign is called an index. For
3
example, the cube root of x is written as √x. Notice that the index
looks similar to an exponent. This is not a coincidence. When we
simplified a square root, we were looking for the base of number
squared or to the second power. When simplifying a higher root, we are
looking for the base of the number raised to the index. The procedure
used to simplify is the same as used for square roots.
3
3
3
Example 1: Simplify the following: √27 , −√27 , and √−27
3
3
3
√27 = √3 ∙ 3 ∙ 3 = √(3)3 = 3
3
3
3
−√27 = − √3 ∙ 3 ∙ 3 = −√(3)3 = -3
3
3
3
√−27 = √(−3) ∙ (−3) ∙ (−3) = √(−3)3 = -3
5
5
5
Example 2: Simplify the following: √32 , − √32 , and √−32
5
5
5
√32 = √2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 = √(2)5 = 2
5
5
5
− √32 = −√2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 = −√(2)5 = -2
5
5
3
√−32 = √(−2) ∙ (−2) ∙ (−2) ∙ (−2) ∙ (−2) = √(−2)5 = -2
Notice from the above examples that a negative radicand results in a
negative answer. This will always be true as along as the index is odd.
However, if the index is even, there is not a number times itself an even
number of times that will result in a negative number. A negative
radicand will not work for an even index and is not a Real number.
A negative outside the radical, odd or even index, simply indicates a
negative answer.
4
4
4
Example 3: Simplify the following: √625 , −√625 , and √−625
4
4
4
√625 = √5 ∙ 5 ∙ 5 ∙ 5 = √(5)4 = 5
4
4
4
−√625 = −√5 ∙ 5 ∙ 5 ∙ 5 = −√(5)4 = -5
4
√−625 = not a Real number
Example 4: Simplify:
4
−5 √16
4
−5√2 ∙ 2 ∙ 2 ∙ 2
4
−5√(2)4
−5 ∙ 2
- 10
Factor the radicand
Rewrite in exponential form
Variables are easier than known values. Variable terms don’t need to be
factored since they are already in exponential form. We use exponent
rules to rewrite the variable terms so that the inside exponent matches
the index. The answer is then the base.
Example 5: Simplify:
6
√a12 b 6
6
√(a2 )6 b 6
a2 b
Rewrite the radicand to have an exponent 6
Notice on the above example, we also could have simply divided the
original exponents by the index.
6
√a12 b 6
a12⁄6 b 6⁄6
a2 b
Divide the exponents by the index
Not all radicands are perfect squares, perfect cubes, etc. As with square
roots, numbers and/or terms that are not part of that perfect radicand are
left inside the radical sign as the simplified radicand.
Example 6: Simplify:
3
√24
3
√2 ∙ 2 ∙ 2 ∙ 3
√23 ∙ 3
3
2 √3
3
Example 7: Simplify:
Factor the radicand
Write in exponential form
4
−3√48x 8 y 9
4
−3√2 ∙ 2 ∙ 2 ∙ 2 ∙ 3 ∙ x 8 ∙ y 9
4
−3√24 ∙ x 8 ∙ y 8 ∙ y1
−3 ∙ 2 ∙ x 8⁄2 ∙ y 8⁄2 √3 ∙ y1
4
−6x 4 y 4 4√3y
Factor the radicand
Rewrite in exponential form