Transcript Unit 6 Notes

Unit 6: Square Roots, Factorials and
Permutations
Section 1: Introduction to Square Roots
• If A = s², then s is the square root of A
• You need to know at least the first ten perfect
squares and their square roots
• Ex1. Name the first 10 perfect squares
• All non-negative numbers have real square
roots, but many of them are irrational
• Numbers have 2 square roots, a positive and a
negative
• Only give the positive unless you are asked for all
• Ex2. Give all square roots of 156.25
• Ex3. If the area of a square is 9.2416 m², what is
the length of each side?
• Radical signs work like a grouping symbol, you are
to complete anything within them before taking
the square root
• Ex4. Solve. Round to the nearest hundredth
12  3
2
2
• Ex5. Estimate between which two whole
numbers the following square root will lie
76
• You can multiply numbers that are under square
roots together
• If you multiply a nonnegative square root by
itself, you are left with the integer within the
square root
• i.e. 7  7  7
• Ex6. Solve and round to the nearest tenth
15  2  6
2
• If you are asked to find an exact answer and the
square root would be irrational, leave the answer
in radical form
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Ex7. Give the exact answer to Ex6.
This skill is imperative for further math classes
Ex8. Solve n² = 361
Ex9. Solve (x + 5)² = 36
Whenever a variable is squared, there will be
TWO answers
• Sections of the book to read: 1-6 and 9-7
Section 2: Simplifying Square Roots
• To extend the practice of leaving answers in exact
form, you can (and must from here on out)
simplify radicals and leave them in exact form
(unless specified otherwise)
• The Product of Square Roots Property states that
you can multiply square roots together, but you
can also use this property in reverse
• Ex1. Multiply 3 5  2 7
• Ex2. Simplify and then multiply 9  5
• In Ex2. you simplified first, but if you have
multiplied first and then simplified, that is known
as simplifying the radical
• Determine the largest perfect square that divides
evenly into the number in the question
• Factor that number out and then find its square
root
• Leave the other number under the radical
• Ex3. Simplify each radical
a) 60
b) 700
c) 2 22  5 14
• Ex4. Simplify
8  72
2
• Ex5. Multiply and then simplify
• Simplify each radical expression
2 2
• Ex6.
24x y z
• Ex7.
2 3 5
300a b c
15  175
5
• Ex8.
• Section of the book to read: 9-7
6x  2x
Section 3: The Multiplication Counting
Principle
• The Multiplication Counting Principle is used to
determine how many possibilities you have when
making multiple choices (one after the other)
• You can only use this principle when your first
choice has no affect on the following choices
• Open your book to page 119, we will be looking
at the stadium pictured at the top
• Ex1. In how many ways can you enter through a
south gate and then exit through a north gate?
• You can use the Multiplication Counting Principle
for more than just two choices
• Ex2. Mr. Halladay writes a test for his history
class. Each of the first 5 questions is multiple
choice with 4 choices. The last 6 questions are
true-false questions. How many ways can a
person answer these questions?
• Ex3. What is the probability that someone who is
randomly guessing on Mr. Halladay’s test will get
all questions correct (as a fraction)?
• Ex4. A license plate has 4 letters followed by 3
numbers. How many license plates are possible?
• Ex5. What if you could not use the letters I and O
in Ex4. How many license plates are now
possible?
• You can use tree diagrams to visually display
possibilities (see page 120)
• Ex6. A test has 10 questions, each with x number
of choices and it has y true-false questions, how
many possible ways can you answer this test?
• Section of the book to read: 2-9
Section 4: Factorials and Permutations
• Factorials are used when you are making choices,
similar to the Multiplication Counting Principle,
but now you CANNOT reuse any options
• The symbol for factorial is !
• 5! = 5 · 4 · 3 · 2 · 1
• Factorials multiply each number (starting with the
given number), decreasing by 1 at a time, until
you get to the number 1
• Most calculators have a factorial button (locate
yours now)
• When you cannot reuse options, each way of
ordering elements is called a permutation
• Ex1. A soccer coach has to write down the name
of all 11 of her players. In how many ways can
she do this?
15!
• Ex2. Evaluate 12!
• Sometimes numbers are too large for calculators
to evaluate and you have to use the definition of
factorial to evaluate
250!
• Ex3. Evaluate 248!
• Section of the book to read: 2-10
Section 5: Exponents and Radicals
• Exponents are not always integers, they can be
rational numbers (fractions and decimals)
• These rational exponents have radical equivalents
1
• i.e. x  x 2
• All square roots are values raised to the .5 power
• Square roots are not the only type of roots
• Cube roots look like this: 3 x
• A cube root is asking you to find what number to
the third power = the number under the radical
symbol
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Ex1. Find 3 8
Calculators do have a button for other roots
3
Ex2. Find 1331
Higher power roots also exist, but they do not
have special names
Find each root
4
10
7
2401
279936
Ex3.
Ex4.
Ex5. 1024
Write each root with a rational exponent
5
7
m
Ex6.
Ex7. ab
Write each term as a root
1
1
Ex8. c 6
Ex9. z 8
Section 6: Rational Exponents
• Rational exponents do not always have a
numerator of 1
• With rational exponents, the numerator is the
exponents and the denominator is the root
2
• i.e. x 3  3 x 2
• If you are solving these with numbers, you can
either find the root first or the exponent first (it
does not matter)
5 8
• Ex1. Write with a fractional exponent n
• Leave the fractions improper (where appropriate)
not as mixed numbers
7
9
v
• Ex2. Write as a radical expression:
• Solve each question. Show steps.
5
• Ex3. 4 2
Ex4. 3 82
4
3
5
• Ex5. 81
Ex6. 125
• Since it doesn’t matter which is done first
(exponent or radical), this equivalence is true:
4
 x
a
b
 x
a
b