Download Algebra 1 Notes SOL A.3 (11.2) Radicals Mrs. Grieser Name: Block

Algebra 1 Notes SOL A.3 (11.2) Radicals
Mrs. Grieser
Name: _______________________________ Block: _______ Date: _____________
Radical Review
y means the xth root of y. The
symbol is called a radical.

x

When x is not specified, we take the 2nd root, or the square root, of the number inside
the radical (the radicand). A square root is a number multiplied by itself (squared) that
gives us the radicand. Example: 4 = 2

A perfect square is a number whose square root is an integer.

We can find other roots. To find the 3rd root, or the cube root, we want to find a number
that when used as a factor 3 times (cubed) gives us the radicand. Example: 3 8 = 2

Calculators can help us find roots of numbers. We can also estimate roots.
o Example: estimate 85
 We know 85 lies between 81 and 100 , but is closer to
85 as being a little more than 9.
81 , so we can estimate
Simplifying Radicals
A radical expression is in simplest form if:



No perfect squares other than 1 are in the radicand
No fractions are in the radicand
No radicals appear in the denominator of a fraction
We use the Product Property of Radicals and Quotient Property of Radicals to simplify:
Quotient Property of Radicals
The square root of a quotient equals the
quotient of the square roots of the
a
a
numerator and denominator:

b
b
Product Property of Radicals
The square root of a product equals the
product of the square roots of the
factors:
ab  a  b
Write the radicand as the product of perfect squares, and then take the roots of those
perfect squares.
Examples for Product Property of Radicals:
a) Simplify
32 =
32
16  2 =
b) Simplify
16 ∙ 2 = 4 2
9x 3 =
9x 3
9  x2  x =
You try: Simplify the radical expressions…
a) Simplify
24
b) Simplify
1
25x 2
9 ∙ x 2 ∙ x = 3x x
Algebra 1 Notes SOL A.3 (11.2) Radicals
Mrs. Grieser
Examples for Quotient Property of Radicals:
a) Simplify
13
=
100
13
100
=
13
10
b) Simplify
1
=
y2
1
y
2
=
1
y
Multiplying Radicals
We can use the product property of to multiply radicals.
Examples:
a)
6∙ 6
= 36
=6
(makes sense!)
Note: When we solve
b) 3x ∙4 x
= 4 3x  x
= 4 3x
= 4x 3
c)
7xy 2 ∙3 x
= 3 7x 2 y 2
2
= 3xy 7
x 2 , we always take the positive value of x.
Rationalizing Denominators
By convention, an expression is not simplified if there is a radical expression in the denominator.
The process of eliminating a radical in a denominator is called rationalizing the denominator.
Examples: Rationalize the denominator…
a)
=
=
5
7
5
7
b)
∙
7
=
7
4
2x
4
2x
2x
∙
2x
=
4 2x
2 2x
=
2x
x
5 7
7
You try: Simplify the radical expressions…
a)
20
d)
96x 2
g)
1
x
b)
72
c)
e)
9x
16
f)
h)
2x 2
5
i)
2
32x 5
4
3
8
3n 2