Download Aim #16: How do we simplify radicals, and rationalize the

CC Geometry H
Aim #16: How do we simplify radicals, and rationalize the denominator of a
fraction?
Do Now:
1. Simplify:
a.
b.
c.
d.
2. Circle the radical expressions that are irrational:
A square root of a number may be negative, however the square root a number
always refers to the principle square root or the positive square root of a number.
i.e. -2 is a square root of 4, but THE square root of 4 is 2.
Definition of the square root: If x ≥ 0, then
is the non-negative number p so
2
that p = x . This definition gives us four facts:
Fact 1:
= a if a ≥ 0. Ex:
= 12
Fact 2:
= -a if a < 0. (the "opposite" of a, not negative a). Ex:
Fact 3:
= |a| for all real numbers a. Ex:
Fact 4:
=
n
= |13|,
n
= a where a is nonnegative. Ex:
1) Simplify as much as possible:
2) Compare the value of
to the value of
**For non-negative real numbers a and b,
.
= 12
= |-13|
To simplify a radical means to first express the radicand as a product -- one
factor a perfect square-- and then simplify.
3) Express each radical in simplest radical form:
b.
a.
c.
d.
Simplifying a radical containing variables
Case 1: even exponents
Copy the base, ___________the exponent by ___.
4) Simplify.
a.
b.
c.
d.
e.
f.
Case 2: odd exponents
5) Simplify.
a.
b.
c.
3) Compare the value of
to the value of
**For nonnegative real numbers a and b, when b ≠ 0,
.
.
To express a number in simplest radical form, there cannot be a radical in the
denominator. If there is a radical in the denominator, we must rationalize it!
**To rationalize (make rational) the denominator, it is sometimes necessary to
multiply the numerator and denominator by a factor that makes the denominator a
perfect square.**
Example 1:
Example 2:
Example 3: Marcus found the scale factor of a dilation to be
. When he
compared his answer to Yelena's, which was
, he told her that one of them
must have made a mistake. Prove that their scale factors are equivalent.
Example 4:
Example 5:
Example 6:
Mixed Practice
Simplify each expression as much as possible and rationalize denominators when
applicable.
1.
4.
2.
3.
6.
7.
8.
10.
11.
13.
14.
9.
12.
15.
17.
19. Find, in simplest radical form, the length of the shorter leg of a right triangle
if the longer leg is 25 and the hypotenuse is 30.
Let's Sum it Up!
To "rationalize a denominator" means that a fractional expression must have a
rational number in the denominator.
Name_______________________
Date: ______________
CC Geometry 10H
HW #16
Express each number in simplest radical form, and rationalize denominators
where applicable.
4.
5.
8.
9. Prove that
are equivalent.
2
10. The area of a square is 54 cm . Express the length of one side, in simplest
radical form.
11. Determine the exact area of the shaded region below:
12. Find, in simplest radical form, the length of the hypotenuse of a right triangle
if the lengths of the two legs are 20 and 12 ft.
Review:
1) In ΔSUN and ΔRAY,
. Which pair of corresponding angles would be
needed to be congruent to prove the triangles similar?
2) Find the value of x:
5x + 2
<
28
<
<
8
16