Transcript Operations With Radical Expressions

Operations With Radical
Expressions
Section 10-3
Goals
Goal
• To simplify sums and
differences of radical
expressions.
• To simplify products and
quotients of radical
expressions.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to solve
simple problems.
Level 4 – Use the goals to solve
more advanced problems.
Level 5 – Adapts and applies
the goals to different and more
complex problems.
Vocabulary
• Like Radicals
• Unlike Radicals
• Conjugates
Like Radicals
Square-root expressions with the same
radicand are examples of like radicals.
Like Radicals
Like radicals can be combined by adding or
subtracting. You can use the Distributive
Property to show how this is done:
Notice that you can combine like radicals by adding or
subtracting the numbers multiplied by the radical and
keeping the radical the same.
Helpful Hint
Combining like radicals is similar to combining
like terms.
Example:
Add or subtract.
A.
The terms are like radicals.
B.
The terms are unlike radicals. Do not
combine.
Example:
Add or subtract.
C.
the terms are
like radicals.
D.
Identify like radicals.
Combine like radicals.
Your Turn:
Add or subtract.
a.
The terms are like radicals.
b.
The terms are like radicals.
Your Turn:
Add or subtract.
c.
The terms are like radicals.
d.
The terms are like
radicals.
Combine like radicals.
More on Like Radicals
• Sometimes radicals do not appear to be
like until they are simplified.
• Simplify all radicals in an expression
before trying to identify like radicals
and combining like radicals.
Example:
Simplify each expression.
Factor the radicands using perfect squares.
Product Property of Square Roots.
Simplify.
Combine like radicals.
Example:
Simplify each expression.
Factor the radicands using perfect squares.
Product Property of Square Roots.
Simplify.
The terms are unlike radicals. Do not combine.
Example:
Simplify each expression.
Factor the radicands using
perfect squares.
Product Property of Square
Roots.
Simplify.
Combine like radicals.
Your Turn:
Simplify each expression.
Factor the radicands using perfect squares.
Product Property of Square Roots.
Simplify.
Combine like radicals.
Your Turn:
Simplify each expression.
Factor the radicands using perfect squares.
Product Property of Square Roots.
Simplify.
The terms are unlike radicals. Do not combine.
Your Turn:
Simplify each expression.
Factor the radicands using perfect squares.
Product Property of Square Roots.
Simplify.
Combine like radicals.
Rationalizing Denominators
Containing Two Terms
• If the denominator has two terms with one or more
square roots, we can rationalize the denominator
by multiplying the numerator and denominator by
the conjugate of the denominator.
• A + B and A – B are conjugates. The product
(A + B)(A – B) = A2 – B2
Example:
12
Rationalize the denominator:
7
3
The conjugate of the denominator is 7  3 . If we multiply
the numerator and the denominator by 7  3 , the simplified
denominator will not contain a radical. Therefore, we
multiply by 1, choosing
12
7

3
12
7

3
7
3
7
3
7
3
7
3
for 1.
Multiply by 1.
Example: Continued
12
7
12

3

7
3
7
3

 7   3
12  7  3 


12

3
7
7
3
2
2
73

12

7
Multiply by 1.
3

3

 A  B  A  B  
A B
2
2
Evaluate the exponents.
Subtract.
4
3

12

7
4
1
Divide the numerator and
denominator by 4.
Example: Continued
3

7

3 or 3 7  3 3
Simplify.
On Rationalizing the Denominator…
If the denominator is a single square root term with nth root:
Multiply numerator and denominator by that square root
expression.
If the denominator contains two terms involving square roots:
Rationalize the denominator by multiplying the numerator and the
denominator bythe conjugate of the denominator.
Example:
Rationalize the denominator:





5
3
3

7  3 
5(3 
(3 
7 

7 
5
3
7
Multiply numerator and
denominator by the conjugate.
7)
7 )( 3 
7)
15  5 7
3  ( 7)
2
2
(A+B)(A-B) = A2 – B2
15  5 7
97
15  5 7
2
Simplify.
Your Turn:
Rationalize the denominator:
5
1 3

55 3
2
1 3
1 3



1 3
5 1 3

1 3 1 3 
55 3
2
5

Your Turn:
Simplify each expression.
1.
2.
6
7

5
1
9

6 7  5
 7  5  7  5 

7


1 9 
9 
7
7
 9 


7

 42  6 5

49  5

9 
7
81  7
 42  6 5
44

9 
74
7

 21  3 5
22
Assignment
• 10-3 Exercises Pg. 621 - 622: #10 – 44 even