Intermediate Algebra Skills, Version 1, Question 19
Rationalize the denominator:
2
2− 3
.
When any number is multiplied by 1, the
value of the number remains the same. We
will use this important principle to express
2
the value
in a different form that
2− 3
retains the same value.
Notice that the denominator is a binomial
expression. To rationalize this expression,
we will multiply the expression by its
conjugate. The conjugate uses the same two
terms, but changes the sign between them.
When we multiply an expression by its
conjugate, we end up with the difference of
the squares of the terms. Any number
(except 0) divided by itself is equal to 1, so
we will multiply the original number
2+ 3
times
. This will rationalize the
2+ 3
denominator without changing the value of
the number.
The answer is 2 2 + 6 .
Expression Conjugate
x+2
x−2
a+b
a−b
2+ 3
2− 3
2+ 3
2+ 3
2
2− 3
=1
×
2+ 3
2+ 3
2 2+ 6
4−3
2 2+ 6
=
1
=2 2+ 6
=
=
2 2+ 2× 3
(2)
2
( )
− 3
2