Download Math 10- Chapter 2 Review

Math 10- Chapter 2 Review
[By Christy Chan, Irene Xu, and Henry Luan]
Knowledge required for understanding this chapter:
1. Simple calculation skills: addition, subtraction, multiplication, and division
2. Understanding of the process for fraction calculations
**2.1: Number System
•
Natural numbers: positive whole numbers used for the purpose of counting
(1,2,3, etc.)
•
Whole numbers: same as natural numbers, but includes 0 (0,1,2,3..)
•
Integers: includes positive & negative whole numbers and 0 (-1,0,1,2,)
•
Rational numbers: all numbers that can be written as a fraction (2, 2/3, -3.5) •
Irrational numbers: all numbers that can’t be written as a fraction, a fraction, a terminating or repeating decimal (√2, π, 3.53319….)
•
Real numbers: include all rational and irrational numbers Æ everything
***2.2 : Greatest common factor & least common multiple
•
A factor of a number is a whole number that when multiplied by another whole
number results in the original number
•
Greatest common factor: the largest factor that two or more numbers can be
divided by evenly.
•
Lowest common multiple: the smallest positive integer that is divisible by all the
terms you are trying to find the LCM for. Use prime factors.
***2.3: Squares and Square roots
●
Perfect Square: All numbers that have whole numbers as square roots. (1, 4, 9,
16, 25, 36, 49, 64, 81, 100)
●
Square Root: A square root of a number x is a number y such that y²=x. (1, 2, 3,
4, 5, 6, 7, 8, 9, 10…)
To find the square roots of perfect squares, there are some methods
without a calculator or table:
Method 1: Factor Tree
For whole numbers √x²=√x*x= x
Divide the whole number into many pairs but eventually 2 pairs that have two
same numbers.
Method 2: Continuous Division
Divide a whole number from the smallest prime number since there is a prime again.
●
Perfect Cube: A number which is the cube of an integer. (0, 1, 8, 27, 64, 125,
216, 343, 512, 729, 1000…)
●
Cube Root: A cube root of number x is a number y such that y³=x. (0, 1, 2, 3, 4,
5, 6, 7, 8, 9, 10…)
Methods to find cube roots of whole numbers
Method 1: Factor Tree
For whole numbers, ³√x ³= ³√x*x*x = x
Divide the whole number into many pairs but eventually 2 pairs that have three
same numbers.
Method 2: Continuous Division
Divide a whole number from the smallest prime since there is a cube root
number.
**2.4 Rational and Irrational Numbers
•
Rational Number: A number that can be represented as a fraction. (Either a
perfect square root or a perfect cube root)
(√16, √100, ³√121, ect.)
•
Irrational Number: A number is a non-repeating, non-terminating decimal
value. (Neither a perfect square root or a perfect cube root)
(π, 0.12123123412345…, √3, ³√4, ect.)
Approximating Irrational Numbers without a Calculator
An approximating of an nth root can be found by determining where the value lies
on a number line.
(Note: Two irrational numbers such as√3 and√300 have the same numerals, but
different decimal point answers because√100 can be represented by a integer, as well
as two irrational cube roots.)
To find closer number to the actual value:
1. Find consecutive integers
2. Figure out how many units from the consecutive integers to that number.
3. By ratios
4. Add the consecutive integer and the unit
***2.5. The Product Rule for Square Roots
The product rule is used when there is a perfect square as a factor.
The product rule is used when there is a perfect cube as a factor.
Entire Radical
An expression such as
is called a mixed radical, and the expression
is
called an entire radical. Both expressions have the same value. Any mixed radical can
be changed to an entire radical.
***2.6. Exponential Notation
*** An exponent tells how many times the base is used as a factor.
Ex. The exponent in this example is 4; the base “2” is repeatedly used 4 times as a
factor(4*4*4*4).
*** What happens if the exponent is 0 or 1?
Exponents of 0 and 1
●
●
●
RULES WHEN DEALING WITH EXPONENTIAL CACULATION:
The Product Rule: ()
For any numbers a and b with exponents m and n:
The Quotient Rule: ()
For any number a with exponents m and n:
The Power Rule
For any numbers a and b with exponents m and n:
Raising a Product to a Power
For any numbers a and b with exponent n:
Negative Exponents
For any number a, a can’t be equal to 0, with exponent n:
Raising a Fraction to Power
For any numbers a and b, b can’t be 0, with exponent n:
Changing from Negative to Positive Exponents
For any non-zero numbers a and b, with exponents m and n:
EXPLANATION TO THE RULE ”Changing from negative to positive exponents”. [Use Rule
”Raising fraction to a power” to simplify:1) to 2) and then to 3).]
1)
2)
3)
Rational Exponents:
For any non-negative real number a and any positive integer n.
Rational Exponents:
For any non-negative real number a and any positive integer n.
[Check page textbook pg.101 for detailed examples]
Chapter 2- Question Sheet
1. Which number systems do the following numbers belong?
(-2, 0, 4,
, √2, π)
2. Find the GCF of 1386 and 1008
3. Find the LCM of 138 and 92
4.
, what is the value of n?
5.
6. Choose a method to solve √1089
7. Choose a method to solve ³√8000
8. Approximate√40 to one decimal place
9. If√2=1.41, √20=4.47, determine the value of√200 and√2000.
10.
Between what two consecutive integers is√216
Chapter 2- Answer Key
1. Real numbers: {4}; whole numbers: {0,4}; integers {‐2,0,4} rational numbers: (‐2, 0, 4, 123) irrational numbers: (√2, π) Real numbers: (‐2, 0, 4,
123, √2, π) 2. 126
(1386=3*3*2*7*11;
1008=2*2*2*2*3*3*7
2*3*3*7=63*2=126)
3. 276
(138=2*3*23; 2*2*23
2*2*3*23=276)
4. n=-1
5.
6.
7.
8.
Let a and b be consecutive integers so that a<√216 < b
Therefore a ²< 216 < b ²
Since 14 ²=196, 15 ²=225, 196<216<225 → 14<√216 <15
So√216 lies between 14 and 15
9.
Let a and b be consecutive integers so that a <√40 < b
Therefore a ²< 40 <b ²
Since 6 ²=36, 7 ²=49, 6<√40 < 7
Then 36<40<49 shows that 40 is 4 units from 36, and 9 units from 49.
By ratios: 4/(4+9)=0.3
6+0.3=6.3
So√40= 6.3
10.
√200=√2*100=√2 *√100=√2 * 10= 1.41*10= 14.1
√2000=√20*100=√20 *√100=√20 * 10=4.47*10= 44.7