SECTION 8.5
Multiplying and Dividing Radical Expressions
Roots
Product Rule
• Ex:
3
32𝑥 4
3
∙ 3𝑥 3
Quotient Rule
• i.e.
If the index is the same, then we can divide
and put the quotient under one radical of the same
index.
The Distance Formula
• The distance formula is
just an application of the
Pythagorean Theorem.
Example 1
Multiply and simplify.
a)
b)
6 3+ 2
5( 72 − 8 )
Example 2
Multiply and simplify.
a)
3+3
b)
13 − 7
5−2
3 + 11
Example 3
Multiply and simplify.
a)
b)
𝑝+5 𝑠
6− 2
2
𝑝−5 𝑠
Example 4
Multiply and simplify.
3
3
9𝑧 − 2 5 9𝑧 + 7
Rationalizing the Denominator
• We don’t want to leave radicals in denominators.
• Thus, we multiply the numerator and denominator
by something that will make the radical go away
in the denominator.
• Ex:
5
3
=
5
3
∙
3
3
=
5 3
5 3
=
3
9
Example 5
Rationalize the denominator.
15
a)
3
−3 2
b)
11
Example 6
Rationalize.
3
𝑥6
2𝑦
Rationalizing the Denominator (cont.)
• If we have a sum or difference involving radicals
in our denominator, we need to multiply the
numerator and denominator by the conjugate.
Expression
5−2 𝑥
5 𝑦+ 𝑧
Conjugate
5+2 𝑥
5 𝑦− 𝑧
Example 7
Rationalize.
4
5+ 6
Example 8
Rationalize.
4 𝑥
𝑥−2 𝑦
Example 9
Write in lowest terms.
6𝑝 + 24𝑝3
3𝑝
SECTION 8.6
Solving Equations with Radicals
Solving an Equation with Radicals
• Ex:
5𝑘 − 3 + 2 = 0
Example 1
Solve.
5𝑘 − 3 + 2 = 0
Example 2
Solve.
7𝑥 − 3 = 6
Example 3
Solve.
5 4𝑥 + 1 = 3 10𝑥 + 25
Example 4
Solve.
𝑥=
𝑥 2 − 4𝑥 − 8
Example 5
Solve.
𝑚2 + 3𝑚 + 12 − 𝑚 − 2 = 0