Download What is the index of a radical?

What is the index of a radical?
The index of a radical is a number which indicates which root is desired of the
number under the radical.
For instance,
3
8 has an index of 3, and means “the cube root of 8”.
Similarly, 4 16 has an index of 4, and means “the 4th root of 16”.
When working with radicals, can the radicand be negative when the index is odd?
Yes.
Can it be negative when the index is even?
No. (unless your class is working with complex numbers.)
Explain the first condition that must be met for a simplified radical.
There are no factors of the value under the radical sign that are perfect squares.
Explain why (5/2^(1/2)) is not simplified and demonstrate the steps we must take to
simplify it.
The expression
5
2
1
2
is not simplified because the denominator contains a term
which includes a radical. The expression is equivalent to
5
.
2
To simplify this expression, multiply the numerator and the denominator by
5
2
5 2
5 2
!
=
=
2
2
2
2 2
2:
Are each of the statements below true or false? Explain your answer without just saying
true or false.
1.) a^(1/2)+b^(1/2)=(a+b)^(1/2). Explain why.
This statement is false. You cannot add terms which have different values under
the radical.
Squaring both sides of this equation gives:
(
a+ b
)
2
2
1$
!
= #( a + b) 2 &
"
%
a a + a b + a b + b b = a+b
a + 2 ab + b = a + b
2 ab = 0
Clearly, for a and b not equal to zero, this last statement will never be true,
therefore the original statement is not a true statement either.
2.) The numerator and denominator of the following must be multiplied by 3^(1/2) to
rationalize 3/(3+3^(1/2)). Explain why.
Multiplying the numerator and denominator by 3^(1/2) will not actually improve
this expression.
In order to simplify this expression, the numerator and the denominator need to be
multiplied by 3 – 3^(1/2):
3
3+ 3
1
2
=
(
)
3 3! 3
3
=
3+ 3 3+ 3 3! 3
(
)(
)
=
9!3 3
9!3 3
=
9!3
9+3 3 !3 3 ! 3 3
=
9 ! 3 3 3 3! 3 3! 3
=
=
6
3(2)
2
(
)
Is this equation true or false? 28-4*2^(1/2) = 24*2^(1/2). Explain why.
1
1
28 ! 4 * 2 2 = 24 * 2 2
28 ! 4 2 = 24 2
28 = 24 2 + 4 2
28 = 28 2
Since this last line is not a true statement, the original equation is false.
The 28 and 4 on the left side cannot be subtracted because they are not “like
terms”. The 4 is attached to the 21/2 terms, and cannot be combined with the 28.
Is the statement below true or false? Explain your answer without just saying true or
false.
1.) When x > 0, is x^(1/3).x^(1/3)=x? Explain why.
No, they are not equal for all x > 0.
When x1/3 and x1/3 are multiplied together the exponents are added:
1
1
x3 ! x3 = x
1 1
+
x3 3 = x
2
3
x =x
This last equation is not a true statement for x > 0 (with the trivial exception of
x = 1)