6-3 Binomial Radical Expressions
Quick Review
Like radicals have the same index and the same radicand. Use the distributive property to add and subtract them. Use the FOIL method to multiply binomial radical expressions. To rationalize a denominator that is a square root binomial, multiply the numerator and denominator by the conjugate of the denominator.
Example
What is the simplified form of
18
+
50
−
8
?
square root of 18 plus square root of 50 minus square root of 8 question mark
18
+
50
−
8
=
3
2
⋅
2
+
5
2
⋅
2
−
2
2
⋅
2
Factor
.
=
3
2
+
5
2
−
2
2
Simplify each radical
.
=
(
3
+
5
−
2
)
2
Combine like terms
.
=
6
2
Simplify
.
table with 5 rows and 2 columns , row1 column 1 , square root of 18 plus square root of 50 minus square root of 8 , column 2 , row2 column 1 , equals . square root of 3 squared , dot 2 end root . plus . square root of 5 squared , dot 2 end root . minus . square root of 2 squared , dot 2 end root , column 2 cap factor , . , row3 column 1 , equals 3 square root of 2 plus 5 square root of 2 minus 2 square root of 2 , column 2 cap simplifyeachradical . . , row4 column 1 , equals . open . 3 plus 5 minus 2 . close . square root of 2 , column 2 cap combineliketerms . . , row5 column 1 , equals 6 square root of 2 , column 2 cap simplify , . , end table
Exercises
Add or subtract if possible.
-
10
27
−
4
12
10 , square root of 27 minus 4 square root of 12
-
3
20
x
+
8
45
x
−
4
5
x
3 , square root of 20 x end root , plus 8 , square root of 45 x end root , minus 4 , square root of 5 x end root
-
54
x
3
3
−
16
x
3
3
cube root of 54 , x cubed end root , . minus . cube root of 16 , x cubed end root ,
Multiply.
-
(
3
+
2
)
(
4
+
2
)
open , 3 plus square root of 2 , close . open , 4 plus square root of 2 , close
-
(
5
+
11
)
(
5
−
11
)
open . square root of 5 plus square root of 11 . close . open . square root of 5 minus square root of 11 . close
-
(
10
+
6
)
(
10
−
3
)
open , 10 plus square root of 6 , close . open , 10 minus square root of 3 , close
Divide. Rationalize all denominators.
-
2
+
5
5
fraction 2 plus square root of 5 , over square root of 5 end fraction
-
3
+
18
1
+
8
fraction 3 plus square root of 18 , over 1 plus square root of 8 end fraction
6-4 Rational Exponents
Quick Review
You can rewrite a radical expression with a rational exponent. By definition, if the nth root of a is a real number and m is an integer, then
a
m
n
=
a
m
n
=
(
a
n
)
m
;
eh super m over n end super , equals , the th , root of eh to the m end root , , equals . open , the th , root of eh , , close to the m . semicolon if m is negative then
a
≠
0
.
eh not equal to 0 . Rational exponents can be used to simplify radical expressions.
Example
Multiply and simplify
x
(
x
3
4
)
.
square root of x . open , the fourth , root of x cubed end root , , close . .
x
(
x
3
4
)
=
x
1
2
⋅
x
3
4
Rewrite with rational exponents
.
=
x
5
4
Combine exponents
.
=
x
5
4
Rewrite as a radical expression
.
table with 3 rows and 3 columns , row1 column 1 , square root of x . open , the fourth , root of x cubed end root , , close , column 2 equals , x super 1 half end super , dot , x super 3 fourths end super , column 3 cap rewritewithrationalexponents . . , row2 column 1 , , column 2 equals , x super 5 fourths end super , column 3 cap combineexponents . . , row3 column 1 , , column 2 equals , the fourth , root of x to the fifth end root , , column 3 cap rewriteasaradicalexpression . . , end table
Exercises
Simplify each expression.
-
25
1
2
25 super and 1 half end super
-
81
1
4
81 super and 1 fourth end super
-
16
1
3
⋅
4
1
3
16 super and 1 third end super , dot , 4 super and 1 third end super
-
5
3
2
⋅
5
1
2
5 super and 3 halves end super , dot , 5 super and 1 half end super
Write each expression in simplest form.
-
(
x
1
4
)
4
open , x super 1 fourth end super , close to the fourth
-
(
−
8
y
9
)
1
3
open . negative 8 , y to the ninth . close super 1 third end super
-
(
9
x
y
2
)
4
open . square root of 9 x , y squared end root . close to the fourth
-
(
x
1
6
y
1
3
)
−
18
open . x super 1 sixth end super . y super 1 third end super . close super negative 18 end super
-
(
x
4
x
−
1
)
−
1
5
open . fraction x to the fourth , over x super negative 1 end super end fraction . close super negative , 1 fifth end super
-
(
x
1
3
y
−
2
3
)
9
open . fraction x super 1 third end super , over y super negative , 2 thirds end super end fraction . close to the ninth
