unit 2 algebra investigations lesson 3: rational and radical expressions notes 3.4: simplify radical...
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Unit 2 Algebra Investigations
Lesson 3: Rational and Radical Expressions
Notes 3.4: Simplify Radical Expressions
Vocabulary
Radical expression: an expression containing a square root symbol.
Radicand: The number or expression inside a radical symbol.
Square Root: The result obtained after taking the square root of a number or expression.
A radical expression is in simplest form if:
1) No perfect square factors (other than 1) are in the radicand:
2) No fractions are in the radicand.
3) No radicals appear in the denominator of the fraction (the bottom part of a fraction).
8 4 2 2 2
5 5
16 4
1 1
24
Product Property of Radicals
The Product Property of Radicals states that the square root of a product equals the product of the square roots.
In other words, =
Examples: 90 9 10 3 10
128 64 2 8 2
Use the product property of radicals to rewrite the following radical expressions in their simplest form:
Checkpoint:
1. 2.
3. 4.
5. 6.
Let’s practice multiplying radicals…
We can work backwards…
• =
and…
2 18 2 18 36 6
5 2 32 5 2 32xy y xy y 2 25 64 5 64xy x y
5 8 40y x y x
Quotient Property
The square root of a quotient (fraction) equals the quotient of the square roots of the numerator and denominator.
Example: a a
b b
What if the denominator (the bottom part of the fraction) isn’t a perfect square???
Example:
In this case, we’ll multiply by the “denominator over the denominator”.
3
5
3 5 15
55 5
The process of eliminating a radical from an expression’s denominator is called rationalizing the denominator.
In order to rationalize this type of denominator, we need to know what a conjugate is:
The binomials a + and a - are called radical conjugates.
b b
3
4
x
x
Let’s look at our previous example:
First we find the conjugate, then we multiply by the “conjugate over the conjugate”…
For this case, the conjugate is
So we can multiply our original expression times the “conjugate over the conjugate.
4 x
Let’s look at another example:
4 2
6
4 2 6
6 6
4 2 6
6 6
24 6 2 4 2
36
x
x
x x
x x
x x
x x
x x x x
x