7.3 Simplified Form for Radicals
7.4 Addition and Subtraction of Radical
Expressions
Professor Tim Busken
Department of Mathematics
Grossmont College
November 5, 2012
Professor Tim Busken
7.3 Simplified Form for Radicals 7.4 Addition and Subtraction of Radic
Simplified Form for Radicals
Learning Objectives:
Write radical expressions in simplified form
Rationalize a denominator that contains only one term
Add or subtract radicals
Professor Tim Busken
7.3 Simplified Form for Radicals 7.4 Addition and Subtraction of Radic
Product Property for Radicals
√
n
ab =
Professor Tim Busken
√
n
a·
√
n
b
7.3 Simplified Form for Radicals 7.4 Addition and Subtraction of Radic
Product Property for Radicals
√
n
ab =
√
n
a·
√
n
b
For example,
√
50 =
√
=
√
25 · 2
25 ·
√
2
Professor Tim Busken
by the product property
7.3 Simplified Form for Radicals 7.4 Addition and Subtraction of Radic
Product Property for Radicals
√
n
ab =
√
n
a·
√
n
b
For example,
√
50 =
√
=
√
=
25 · 2
25 ·
5·
√
√
2
by the product property
2
Professor Tim Busken
7.3 Simplified Form for Radicals 7.4 Addition and Subtraction of Radic
Product Property for Radicals
√
n
ab =
√
n
a·
√
n
b
For example,
√
50 =
√
=
√
=
25 · 2
25 ·
5·
√
√
2
by the product property
2
√
= 5 2
Professor Tim Busken
7.3 Simplified Form for Radicals 7.4 Addition and Subtraction of Radic
Product Property for Radicals
√
n
ab =
√
n
a·
√
n
b
For example,
√
50 =
√
=
√
=
25 · 2
25 ·
5·
√
√
2
by the product property
2
√
= 5 2
There
is no sum properties
of radicals that says
√
√
√
n
n
a +b = na+ b
Professor Tim Busken
7.3 Simplified Form for Radicals 7.4 Addition and Subtraction of Radic
Product Property for Radicals
√
n
ab =
√
n
a·
√
n
b
For example,
√
50 =
√
=
√
=
25 · 2
25 ·
5·
√
√
2
by the product property
2
√
= 5 2
There
is no sum properties
of radicals that says
√
√
√
n
n
a +b = na+ b
If that was
√
√
√
√
√
√ true, then
16 = 4 + 4 + 4 + 4 = 4 + 4 + 4 + 4 = 2 + 2 + 2 + 2 = 8
Professor Tim Busken
7.3 Simplified Form for Radicals 7.4 Addition and Subtraction of Radic
Quotient Property for Radicals
r
n
√n
a
a
= √n
b
b
Professor Tim Busken
(b , 0 )
7.3 Simplified Form for Radicals 7.4 Addition and Subtraction of Radic
Quotient Property for Radicals
r
n
√n
a
a
= √n
b
b
(b , 0 )
For example,
r
√
2
2
= √
25
25
=
√
2
5
Professor Tim Busken
by the quotient property
7.3 Simplified Form for Radicals 7.4 Addition and Subtraction of Radic
When we write radical expressions in simplified form it means we are writing
the expressions so that they are easiest to work with.
Professor Tim Busken
7.3 Simplified Form for Radicals 7.4 Addition and Subtraction of Radic
Simplified Form for Radical Expressions
A radical expression is in simplified form if
1
2
3
None of the factors of the radicand can be written as powers greater
than or equal to the index—that is, no perfect squares can be factors of
the quantity under a square root sign, no perfect cubes can be factors of
what is under a cube root sign, and so forth;
There are no fractions under the radical sign; and
There are no radicals in the denominator.
Professor Tim Busken
7.3 Simplified Form for Radicals 7.4 Addition and Subtraction of Radic
Simplified Form for Radical Expressions
A radical expression is in simplified form if
1
2
3
None of the factors of the radicand can be written as powers greater
than or equal to the index—that is, no perfect squares can be factors of
the quantity under a square root sign, no perfect cubes can be factors of
what is under a cube root sign, and so forth;
There are no fractions under the radical sign; and
There are no radicals in the denominator.
Try these on your own! Write each radical expression in simplified form.
Assume that any variables represent positive quantities.
p
√3
50x 2 y 3
32a 4 b 6
p
3
54x 5 y 8
p
4
80x 3 y 8
√
75m5 n8
Professor Tim Busken
7.3 Simplified Form for Radicals 7.4 Addition and Subtraction of Radic
Rationalizing the Denominator
Try these on your own! Write each radical expression in simplified
form. Assume that any variables represent positive quantities.
r
50
9x 2
r
5
6
r
4
5
−
r
5
2
Professor Tim Busken
7.3 Simplified Form for Radicals 7.4 Addition and Subtraction of Radic
Rationalizing the Denominator
Try these on your own! Write each radical expression in simplified
form. Assume that any variables represent positive quantities.
√
2 3x
5y
√
r
3 5x
2y
12x 5 y 3
5z
7
√3
4
r
5
√3
9
Professor Tim Busken
48x 3 y 4
7z
7.3 Simplified Form for Radicals 7.4 Addition and Subtraction of Radic
Addition or Subtraction of Radical Expressions
We have been adding and subtracting polynomials by combining
like terms. We do the same with radical expressions.
Professor Tim Busken
7.3 Simplified Form for Radicals 7.4 Addition and Subtraction of Radic
Addition or Subtraction of Radical Expressions
Definition
Two radical terms are said to be similar, or like terms if they have
the same index and same radicand.
Professor Tim Busken
7.3 Simplified Form for Radicals 7.4 Addition and Subtraction of Radic
Addition or Subtraction of Radical Expressions
Definition
Two radical terms are said to be similar, or like terms if they have
the same index and same radicand.
Identify whether or not each radical expression contains like
terms or not.
√3
2 5+
√3
√
√
5
√
3+5 3−2 3
√7
2 4+
√7
9
Professor Tim Busken
7.3 Simplified Form for Radicals 7.4 Addition and Subtraction of Radic
Adding or Subtracting Radical Expressions
To add or subtract radical expressions, put each in simplified form and
apply the distributive property, if possible. We can add only like radicals.
We must write each expression in simplified form for radicals before we
can tell if the radicals are similar.
Try these on your own! Write each radical expression in simplified form.
Then combine like radicals. Assume that any variables represent positive
quantities.
√
√
7−3 7
√
√
6x a + 5x a
√6
7 7−
√
√6
√6
7+4 7
√
√
5x 8 + 3 32x 2 − 5 50x 2
Professor Tim Busken
7.3 Simplified Form for Radicals 7.4 Addition and Subtraction of Radic
Adding or Subtracting Radical Expressions
To add or subtract radical expressions, put each in simplified form and
apply the distributive property, if possible. We can add only like radicals.
We must write each expression in simplified form for radicals before we
can tell if the radicals are similar.
Try these on your own! Write each radical expression in simplified form.
Then combine like radicals. Assume that any variables represent positive
quantities.
√3
p
3
2 x 8 y 6 − 3y 2 8x 8
√
√
5a 2 27ab 3 − 6b 12a 5 b
√3
√3
b 24a 5 b + 3a 81a 2 b 4
Professor Tim Busken
7.3 Simplified Form for Radicals 7.4 Addition and Subtraction of Radic