Transcript Simplifying Radicals

Lesson 10-3 Warm-Up
ALGEBRA 1
“Operations With Radical
Expressions” (10-3)
What are “like
and unlike
radicals”?
like radicals: radical expressions that have the same radicand
Example: 4
7 and -12
7 are like radicals.
unlike radicals: radical expressions that do not have the same radicand
Example: 3
11 and 2
5 are NOT like radicals
How can you
combine like
radicals?
You can combine like radicals using the Distributive Property.
Example: Simplify
2 and 3
2.
2 + 3
2 = 1
(1 + 3)
2
4
2
2 + 3
2
Both terms contain
2 .
Use Distributive Property to combine
like terms [like 2x + 3x = (2 + 3)x = 5x]
Simplify.
ALGEBRA 1
Operations With Radical Expressions
LESSON 10-3
Additional Examples
Simplify 4
4
3+
3=4
3+1
= (4 + 1)
=5
3
3+
3
3
3.
Both terms contain
3.
Use the Distributive Property to
combine like radicals.
Simplify.
ALGEBRA 1
Operations With Radical Expressions
LESSON 10-3
Additional Examples
Simplify
8
5–
45 = 8
5–
9•5
=8
5–
9•
=8
5–3
= (8 – 3)
=5
5
5–
8
5
5
45.
9 is a perfect square and a factor of 45.
5
Use the Multiplication Property of
Square Roots.
Simplify
9.
Use the Distributive Property to
combine like terms.
Simplify.
ALGEBRA 1
Operations With Radical Expressions
LESSON 10-3
Additional Examples
Simplify
5(
8 + 9) =
=
=2
5 (
40 + 9
4•
8 + 9).
5
10 + 9
10 + 9
5
Use the Distributive Property.
5
Use the Multiplication Property
of Square Roots.
Simplify.
ALGEBRA 1
“Operations With Radical
Expressions” (10-3)
How do simplify
using FOILing?
If both radical expressions have two terms, you can FOIL in the same way you
would when multiplying two binomials.
Example:
Given.
ALGEBRA 1
Operations With Radical Expressions
LESSON 10-3
Additional Examples
Simplify (
(
6–3
6 – 3 21)( 6 + 21)
= 36 +1 126 - 3 126 – 3
21)(
441
6+
21).
Use FOIL.
=6–2
126 – 3(21)
Combine like radicals and
simplify 36 and 441.
=6–2
9 • 14 – 63
9 is a perfect square factor of 126.
=6–2
9•
Use the Multiplication Property of
Square Roots.
=6–6
14 – 63
= – 57 – 6
14 – 63
14
Simplify
9.
Simplify.
ALGEBRA 1
“Operations With Radical
Expressions” (10-3)
What are
“conjugates”?
conjugates: The sum and the difference of the same two terms.
Example:
Rule: The product of two conjugates is the difference of two squares.
Example:
FOIL
Simplify.
Notice that the product of two conjugates containing radicals has no radicals.
How can we
rationalize a
denominator
using
conjugates?
Recall that a simplified radical expression has no radical in the denominator. If
the denominator does contain a radical, we need to get rid of it through
rationalization. If the denominator is a sum or difference that contains a radical
expression, we can rationalize it by multiplying the numerator and denominator
by the conjugate of the denominator.
Example: To rationalize
, multiply by
ALGEBRA 1
“Operations With Radical
Expressions” (10-3)
Example:
Multiply (the denominator is the sum of
the squares)
Divide 6 and 3 by the common factor 3
Simplify.
ALGEBRA 1
Operations With Radical Expressions
LESSON 10-3
Additional Examples
8
7–
Simplify
8
7–
=
3
7+
7+
•
3
3
3
.
Multiply the numerator and
denominator by the conjugate
of the denominator.
=
8( 7 + 3)
7–3
Multiply in the denominator.
=
8( 7 +
4
Simplify the denominator.
= 2(
7+
=2
7+2
3)
3)
3
Divide 8 and 4 by the common
factor 4.
Simplify the expression.
ALGEBRA 1
Operations With Radical Expressions
LESSON 10-3
Additional Examples
A painting has a length : width ratio approximately equal to
the golden ratio (1 + 5 ) : 2. The length of the painting is 51 in. Find
the exact width of the painting in simplest radical form. Then find the
approximate width to the nearest inch.
Define:
51 = length of painting
x = width of painting
Words: (1 +
5) : 2 = length : width
5) = 51
x
2
Translate: (1 +
x (1 +
5) = 102
x(1 + 5) =
102
(1 + 5)
(1 + 5)
Cross multiply.
Solve for x by dividing both side by (1+
ALGEBRA 1
5).
Operations With Radical Expressions
LESSON 10-3
Additional Examples
(continued)
x=
(1 –
102
•
(1 –
(1 + 5)
x=
102(1 – 5)
1–5
x=
102(1 –
–4
5)
x = – 51(1 –
5)
2
x = 31.51973343
5)
5)
Multiply the numerator and the
denominator by the conjugate
of the denominator.
Multiply in the denominator.
Simplify the denominator.
Divide 102 and –4 by the
common factor –2.
Use a calculator.
x 32
The exact width of the painting is
– 51(1 –
2
5)
inches.
The approximate width of the painting is 32 inches.
ALGEBRA 1
Operations With Radical Expressions
LESSON 10-3
Lesson Quiz
Simplify each expression.
1. 12 16 – 2
40
4. (
16
3–2
21)(
–123 + 3
7
2.
3+3
20 – 4
–2 5
21)
5
5.
3.
16
5–
–8
2( 2 + 3
2+3 6
3)
7
5–8
7
ALGEBRA 1