BGS - 03. Factors, Powers and Roots Flashcards
Factors
Factors of a number are whole numbers that divide into it
Factors of 6;
6 (6x1), 3 (3x2), 2 (2x3), 1 (1x6)
Factors of 8;
8 (8x1), 4 (4x2), 2 (2x4), 1(1x8)
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Highest Common Factor
Highest common factor (HCF) of a pair of numbers is the highest number that is a factor for both numbers
What is the HCF of 24 and 40
Factors of 24;
24, 12, 8, 6, 4, 2, 1
1x24, 2x12 3x8, 4x6, 6x4, 8x3, 12x2
Facotrs of 40
40, 20, 10, 8
1x40, 2x20, 4x10, 8x5
Common Factor: 8
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Least common multiple (LCM)
The lowest common multiple of 2 numbers is the lowest multiple which appears in the multiples of each number.
What is the LCM of 4 and 6
Multiples of 4;
4, 8, 12, 16, 20, 24, 28, 32, 36….
Multiples of 6;
6, 12, 18, 24, 30, 36…
Lowest Common Multiple (LCM) is 12 as it appears in both multiples strings and is the lowest commoon shared number;
LCM = 12
1
Prime Factors
It is possible to write out a number as a long multiplication until such time you reach a prime number (prime number being divisible only by itself and 1).
Example 24 = 2x2x2x3
First we divide the number by 2 to get the simplest form i.e.
24 = 2 x 12
We can further expand this as 2x6 = 12 so…
24 = 2(2 x 6) note it is not written in brackets however.
Becomes..
24 = 2 x 2 x 6
We can do this again as 2 x 3 = 6
Becomes..
24 = 2 x 2 x 2 x 3
We have reached a point where 3 is a prime number
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Highest Common Factor - Using Prime Factors
Write down the prime factors for 2 numbers and it is possible to find the Highest common factor by cancelling off of both lists any factors which are not the same in both lists.
Find the HCF for 24 and 36 using a prime factor list
Prime factors:
24 = 2 x 2 x 2 x 3
36 = 2 x 2 x 3 x 3
It can be seen in this example that the 3rd numbers do not match so can be cancelled off. This gives the HCF answer;
2 x 2 x 3 ⋉ 2(2 x 3) ⋉ 2 x 6 = 12
HCF = 12
3
Lowest Common Multiple - Using Prime Factors
Write down the prime factors for 2 numbers and it is possible to find the Lowest Common Multiple by looking at the prime factors and determining the number that appears the most in each string.
Find the LCM of 24 and 36 using a prime factor list
Prime factors:
24 = 2 x 2 x 2 x 3
36 = 2 x 2 x 3 x 3
For 24, 2 appears 3 times
For 36, 3 appears 2 times
24 = (2 x 2 x 2) x 3
36 = 2 x 2 x (3 x 3)
Becomes;
(2 x 2 x 2) x (3 x 3) ⋉ 8 x 9 = 72
LCM = 72
HOWEVER - in this example, only the numbers 2 and 3 appear. If different Prime factors appear, even if just once, and that number does not appear in the prime factor list of the other number, this must be included in the LCM multiplication.
Example;
90 = 2 x 3 x 3 x 5
175 = 5 x 5 x 7
Becomes;
90 = (2) x (3 x 3) x (5)
175 = (5 x 5) x (7)
As 5 appears more times in 175, we drop 5 from 90. However 2 only appears in 90, and 7 only appears in 175 so these must be included
Becomes;
2 x (3 x 3) X (5 x 5) x 7
(2 x 9) x (25 x 7)
18 x 175
LCM = 3150
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Squares, Cubes and Roots
Where numbers are raised to a power and you are asked to multiply them together, ADD the powers together
2^2 x 2^3
2 + 3 = 5
Becomes
2^5
where ^ means the following number is subscript i.e. 2^2 is 2²
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Squares, Cubes and Roots
Where find the power of a number already raised to a power, the powers need to be multiplied together
(3^3)²
This is the same as saying,
(3 x 3 x 3)²
Which is now the same as saying;
(3 x 3 x 3) x (3 x 3 x 3)
which is the same as saying
3^6
(3 x 3 x 3 x 3 x 3 x 3)
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Standard Index Form
A useful way of writing large numbers.
Typically written as
Number x 10^n
The power to n means the number of places the decimal has moved.
Where positive n means the decimal moves right,
negative n means the decimal moves left
1.5 x 10^8
Move decimal place 8 times to the right
150,000,000.
Or;
1.5 x 10^-3
Move decimal place 3 times to the left
0.0015
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Adding/Subtracting Standard Form
When adding or subtracting standard form, convert the equation to ordinary numbers, perform the calculation, then convert back to standard form
(4.59 x 10^5 ) + (2.1 x 10^4)
⋉
459,000 + 21,000
= 480,000
Convert 480,000 back to standard form
Becomes…
4.8 x 10^5
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Multiply/Divide Standard Index Form
When multiplying by powers, you ADD the powers together
When you dividing by powers, you SUBTRACT the powers
Multiply;
10^5 x 10^3
5 + 3 = 8
Becomes…
10^8
Divide;
10^5 / 10^3
5 - 3 = 2
Becomes…
10^2
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Multiply/Divide Standard Index Form
When given an equation where you must multiple 2 multiplications of powers, use the same rule where;
When multiplying by powers, you ADD the powers together
When you dividing by powers, you SUBTRACT the powers
But where there is a multiple between the 2 brackets you multiple the none powered numbers, and where there is a divide between the 2 brackets, you divide the none powered numbers
Examples;
(1.28 x 10^6) x (4.5 x 10^3)
(8.22 x 10^8) / (2.0 x 10^4)
Multiple
(1.28 x 10^6) x (4.5 x 10^3)
1.28 x 4.5 = 5.76
6 + 3 = 9
Becomes..
5.76 x 10^9
Divide
(8.22 x 10^8) / (2.0 x 10^4)
8.22 / 2.0 = 4.11
8 - 4 = 4
Becomes…
4.11 x 10^4
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Zero, Negative and Fractional Powers
Any number to the power of zero i.e. n^0 is always 1
A number powered to ½ is the same as the square root
A number that is powered to ⅓ is the same as the cubed root
Power to half
4^½ ⋉ √4
= 2
i.e. 2 x 2 = 4
Power to third
27^ ⅓ ⋉ ^3√27
= 3
i.e. 3 x 3 X 3
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Minus Power
When you have a number powered by a negative number, this is the same as saying 1 divided by the number to the power it is raised for.
What is 4^-3 in standard index form
4^-3 ⋉ 1 / 4^3
Note that we have put a 1 over the 4, and changed the negative power to a positive power
1 / 4^3 = 1 / 64
= 0.015625
Written as standard index form;
**1.5625 x 10^-2 **
