**Chapter 7: Section 1 Radical Expressions**

Every positive real
number has 2 square roots–one that is positive

(which is called the principal square root)of the number and one

that is negative.

EX1. What are the square roots of 25? What is the
principal square

root of 25?

EX2. What are the square roots of 10? What is the
principal square

root of 10?

The square roots of
negative real numbers are complex (not real)

numbers.

EX3. Explain why is not a real number.

If a > 0, then denotes the principal square root of a.

EX4.

EX5.

EX6.

(Be careful when
taking the square root of variables unless you are

told that the variables represent positive real numbers.)

for all real numbers x.

If x > 0, then but if x < 0, the
. To ensure

that is never negative, we use |x|.

EX7.

EX8.

The cube root of a
number a, denoted by is a number b such

that

EX9. Explain why 2 is a cube root of 8 but 4 is not.

for all real numbers. [Why don’t we use
absolute value

bars around x here like we did for square roots?]

EX10.

In general, the nth
root of a, denoted by , is a number b such

that . [The nth root of a may not always be a
real number.]

EX11.

EX12.

EX13.

EX14.

The graph of the square root function is shown below.

What is the domain of this function?

What is the range of this function?

EX15. Graph and find its domain and range.

EX16. Graph and find its domain and range.

EX17. Graph and find its domain and range.

EX18. Graph and find its domain and range.

EX19. Graph and find its domain and range.

EX20. The height of a cylinder is given by the formula:

where V represents the volume of the cylinder and r is its radius.

Find the height of a cylinder whose radius is 3 cm and
whose

volume is 16π cubic cm.

Find the volume of a cylinder whose height is 5 inches and
whose

radius is 2 inches.

**Chapter 7: Section 2 Application of Radicals**

**Pythagoras**

1) Born about 575 B.C. on the Greek island of

Samos.

2) The first man to call himself a philosopher–a lover of

wisdom.

3) Founded the Pythagoreans in Croton in southern Italy.

4) The Pythagoreans believed in metempsychosis–a belief

that the soul in reincarnated into the bodies of humans,

animals, and vegetables.

5) Although the Pythagoreans were vegetarians, they didn’t

believe in eating beans. One possible explanation for this

is they believed that every time you farted part of your soul

was lost.

6) Died about 435 B.C.

The Pythagorean Theorem states:
_______________________________

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EX1. Find the from one corner of a room to the diagonally
opposite

corner if the room is rectangular in shape and is 15 ft by 22 ft.

EX2. A ladder that is 17 feet long is leaning against a
house. If the

ladder is placed 7 feet from the base of the house, how far up on

the side of the house does the ladder reach?

EX3. How are from the base of a house should you put a 22
ft ladder

if you want to reach a window that is 16 feet above the ground?

EX4. A baseball field is in the shape of a diamond
(square) that is 90

ft on each side. How far is it from home plate to second base?

One of the many proofs of the Pythagorean Theorem was
discovered by

James A. Garfield, the twentieth president of the United States.

1) Born on November 19, 1831. Last president to be born in

a log cabin.

2) In 1857 was named president of Hiram College in Ohio.

3) In 1858, he married Lucretia Randolph. They had 7

children.

4) Served as a major general in the Civil War.

5) In 1876, he discovered an original proof of the
Pythagorean

theorem.

6) Was elected the twentieth president of the United
States

of American in 1880.

7) Shot twice on July 2, 1881. Died on September 19, 1881.

The converse of the Pythagorean Theorem states:
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EX5. If the lengths of the sides of a triangle are 5 cm,
12 cm, and 13

cm is the triangle a right triangle? Justify your answer.

EX6. If the lengths of the sides of a triangle are
ft, ft,
and

7 ft is the triangle a right triangle? Justify your answer.

EX7. Suppose that x and x+3 are two sides in a right
triangle whose

hypotenuse is 15 m. Find the lengths of these two sides.

Let
and be
the coordinates of two points. The

distance between these points is given by the formula:

EX8. Find the distance between the following pairs of points.

(3, 4) and (2, 11)

(-1, -6) and (-4, -9)

A scalene triangle is
____________________________________

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An isosceles triangle
is __________________________________

_____________________________________________________.

An equilateral
triangle is ________________________________

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EX9. Find the length of each side of the triangle whose
vertices are

(-2, 1), (-1,5), and (4,2). Classify this triangle as either scalene,

isosceles, or equilateral.