Olympiads School MCu9 x16 class 1: handout
First name: ______________ Last name: ________________
Chapter 1 Rational Numbers and Exponents
1. Rational Numbers
Number Systems:
Natural numbers, N = {
Whole numbers, W = {
Integers, I = {
}
}
}
2. Rational Number Operation
Recall: BEDMAS – Brackets, exponents, division/multiplication, addition/subtraction
Example 1: Evaluate without using a calculator. Show your work.
a)
1 3 5
+ ×−
2 4 6
3 1
b) −3 + −
4 2
3. Exponents and Square Root
1
2
2 1 7
c) 1 + 2 ÷
3 5 10
Olympiads School MCu9 x16 class 1: handout
This mark
is called the radical sign
The number under the radical sign is called the radicand.
Example: Evaluate.
1
a)
4
e)
3
36
h) 2 ⋅ 3 ⋅ 5 ⋅ 2 ⋅ 3 ⋅ 5
2
2
d) − −
3
2
b) (-6)
c) -6
f) 13 ⋅13
g)
i) √𝑎𝑎6
j) √𝑎𝑎4 𝑏𝑏 8 𝑐𝑐12
4
−13 ⋅ (−13)
4. The Exponent Rules
1) Product Rule
Example: Simplify the expression
a) y 4 ⋅ y 7
b) (−2a 3b 2 )(4a 5b3 )
2) Quotient Rule
Example: Simplify the expressions.
−25a 7b 4
y10
a) 3
b)
5a 2b
y
3) Zero Exponent
Example: Evaluate
a) (2 xyz )0
b) 3x0
4) Power Rule
Example: Simplify the expression ( y 4 ) 2
5) Power of a Product Rule
Example: Simplify the expression (ab)5
2
Olympiads School MCu9 x16 class 1: handout
6) Power of a Quotient Rule
2
Example: Simplify the expression
x
7) Negative Exponent Rule
Example: a) 5-2
b) 4x-2
c)
5. Simplifying Exponential Expressions
Example: Simplify
(5a 3 )(2a 4 )
a)
(a 2 )
3 x3 y −2
e) 5 −3
4x y
b)
−3
f)
(3 x 4 y 5 ) 2
x2 y5
c)
(−4 x3 y −1 ) 2 (5 x3 y −2 )0
(2 x 4 y )3
(3 x 4 y −2 ) −3
(2 x 3 y 2 ) −2
g)
6. Scientific Notation
Scientists have developed a shorter method to express very
large numbers. This method is called scientific notation.
Scientific Notation is based on powers of the base number 10.
3
d)
(3 x −5 y 2 )0
(4 x −3 y 2 ) −2
(6 x3 y −2 ) −2 (3 x 4 y −5 ) 2
(2 x 4 y 2 ) −3
5
x −3
y −7
Olympiads School MCu9 x16 class 1: handout
The number 123,000,000,000 in scientific notation is written as: _____________________
The first number 1.23 is called the _____________________. It must be greater than or equal to
______ and less than ________.
The second number is called the ________. It must always be 10 in scientific notation.
For small numbers we use a similar approach. Numbers less than 1 will have a negative
exponent. A millionth of a second is:
0.000001 sec. = ____________________ in scientific notation.
7. Working with Scientific Notations using Exponential Rules
1) Multiply and Divide Two Numbers Written in Scientific Notation.
•
•
•
Multiply/divide the decimal numbers.
Multiply/divide the powers of 10 by adding/subtracting their exponents.
Convert your answer to scientific notation if necessary.
a) (9 × 10-1) × (3 × 1010)
b)
3.5 ×10−6
5 ×10−2
2) Add and Subtract Two Numbers Written in Scientific Notation.
•
•
•
Rewrite the number with the smaller exponent so that it has the same exponent as the
number with the larger exponent by moving the decimal point of its decimal number.
Add/subtract the decimal numbers. The power of 10 will not change.
Convert your result to scientific notation if necessary.
a) 4.9 × 102 + 7.9 × 103
b) 2.45 × 10-4 - 6.1 × 10-5
4
Olympiads School MCu9 x16 class 1: handout
3) Application
Example: At its closest, the planet Neptune is 4,300,000,000 kilometers away from Earth. A
group of astronauts from Earth want to make it to Neptune in 20,000 days. If they travel the
same number of kilometers each day, how many kilometers will they travel each day?
Convert both numbers to scientific notation before solving.
Practice in class
1. Evaluate each expression. Show your work.
1 1
3 −3
2 −2 5
a) −4 + × −3
b) − × − 2 ÷
5 2
5 8
5 3 6
1 5 1
c) −2 ÷ − 1 + 3
4 3 2
2. Simplify.
−32 x5 y 4
a)
(4 xy 2 ) 2
12 x 4 y
b)
−3 xy
2
c)
6𝑥𝑥 −2 𝑦𝑦3 ⋅2−1 𝑥𝑥 −3 𝑦𝑦 −1
(3𝑥𝑥 0 𝑦𝑦 3 )−3
3. Evaluate.
a) (266×10−6) + (8.6×10−6)
b) (7.14×105) − (5.5×104)
c) (2.56×10−3) × (3.8×106)
d) (4.97×108) ÷ (7×105)
5
0
