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Video 1
Video 1
The three rules explained in this video are not hard to understand, it is mostly common sense and if you understood indices part 1, you will understand these rules.
Lets quickly understand them now.
REMEMBER:
5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 = 59 (Just count how many fives there are)
So what does 52 x 54 equal?
What does 52 equal? 5 x 5
What does 54 equal? 5 x 5 x 5 x 5
Then this means
52 x 54 = 5 x 5 x 5 x 5 x 5 x 5 = 56
This shows how we can add the 2 and 4 to get 56 .
BUT REMEMBER, WE ARE ONLY ABLE TO DO THIS BECAUSE THE BASE IS THE SAME. THE BASE IS 5.
e.g. 23 x 53 = 2 x 2 x 2 x 5 x 5 x 5 (WE CANNOT SIMPLIFY HERE LIKE WE DID PREVIOUSLY AS IT DOES NOT MAKE SENSE TO DO SO)
WHY DO WE NEED TO KNOW THIS?
It is sometimes more convenient to simplify things and write them in a shorter way, like explained in the indices part 1 page.
What is faster and easier to write 6 x 63 x 67 OR 611? It is 611
6 = 61 (any number to the power of 1 is just the same number) 63 = 6 x 6 x 667 = 6 x 6 x 6 x 6 x 6 x 6 x 6
This tells us that when we multiply index numbers with the same base we can add the powers of each index number. Then we can write the base to the power of the result of that addition.
e.g. If we need to simplify 52 x 54 , we can first add the powers (2 + 4 = 5) and then write the base (which is 5 in this case) to the power of the result of the addition. Thus we can write 55.
Question 1: Simplify 63/62
63/62 = (62 x 6)/62 = 62/62 x 6 = 1 x 6 = 6
The idea is to write the bigger number (63), which in this case is the numerator, as a product of the smaller number (62), which in this case is the denominator, and the (base)n, remember the base in the above example is 6.
So
63 = 62 x (base)n
The base in our case is 6, so we can write
63 = 62 x (6)n, clearly n = 1, we know this has we can add the powers, as learnt above.
So 63 = 62 x 6
That is how we get
63/62 = (62 x 6)/62 = 62/62 x 6 = 1 x 6 = 6
Question 2: Simplify 75/78
The bigger number is 78 and the smaller number is 75. This is obvious, because the bigger the power, the bigger the number when the base, in this case which is 7, is the same.
Now lets find a way to write the bigger number (78) as a product of the (base)n and the smaller number 75.
78 = 75 x (base)nWe know that the base is 7, so we can write
78 = 75 x 7n
How do we find n?
Given, what we have learned in the multiplication section above, adding 5 and n (which are the powers) needs to equal 8, so n equals 3.
So 78 = 75 x 73
So now we can rewrite the question as
75/78 = 75/(75 x 73) = 75/75 x 1/73 = 1 x 1/73 = 1/73 = 7-3
Being able to manipulate expressions is important in maths, we have purposely focussed on using 75 because you can see above we have been able to get (7^5)/(7^5), this gives us 1 as they cancel each other out.
ANOTHER WAY TO THINK IS DISCUSSED BELOW
Simplify 53/54
53/54 = (5 x 5 x 5) / (5 x 5 x 5 x 5) = 5/5 x 5/5 x 5/5 x 1/5
You can see that we have rewritten (5 x 5 x 5) / (5 x 5 x 5 x 5) as 5/5 x 5/5 x 5/5 x 1/5. Multiply the fractions and see that you will get exactly (5 x 5 x 5) / (5 x 5 x 5 x 5).
5/5 x 5/5 x 5/5 x 1/5 = 1 x 1 x 1 x 1/5 = 1/5 = 5-1
What we are essentially doing is cancelling out the fives; there are three fives being multiplied with each other on the numerator and four fives being multiplied with each other on the denominator.
(5 x 5 x 5) / (5 x 5 x 5 x 5) = 1/5
You can see that the fives in the purple text colour cancel each other out. This leaves us with 1/5 at the end.
Lets try another question, 534/537?
We know that there would 34 fives being multiplied by each other on the numerator, and 37 fives being multiplied by each other on the denominator, the 34 fives out the 37 fives on the denominator and the 34 fives on numerator would cancel each other out, so we are left with 3 fives being multiplied on the denominator. This shows that
534/537 = 5-3 = 1/53 = 1/125
So essentially we can subtract the powers (34-37) to get -3 as the power.
This tells us when we divide index numbers with the same base we can subtract the powers of each index number. Then we can write the base to the power of the result of that subtraction.
e.g. If we need to simplify 53/54, we can first subtract the powers (3 - 4 = -1) and then write the base (which is 5 in this case) to the power of the result of the subtraction. Thus we can write 5-1.
We see that the third rule in the video shows an index number to the power of n. Lets go through the two examples (83)2 and (64)5 .
What does (83)2 mean?
(83)2 = 83 x 83 = 86 (Remember we can add the powers as explained above)
So essentially we can multiply the 3 and 2 together
Does this seem familiar? If you think about, when we write out (83)2 = 83 x 83 = 86 we are really just using the multiplication rule discussed above. In this case though, the base is an index number.
What does (64)5 mean?
(64)5 = 64 x 64 x 64 x 64 x 64
64 x 64 x 64 x 64 x 64 = 620 = 6(4 x 5)
Lets quickly understand them now.
Multiplying
REMEMBER:Multiplying
5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 = 59 (Just count how many fives there are)
So what does 52 x 54 equal?
What does 52 equal? 5 x 5
What does 54 equal? 5 x 5 x 5 x 5
Then this means
52 x 54 = 5 x 5 x 5 x 5 x 5 x 5 = 56
This shows how we can add the 2 and 4 to get 56 .
BUT REMEMBER, WE ARE ONLY ABLE TO DO THIS BECAUSE THE BASE IS THE SAME. THE BASE IS 5.
e.g. 23 x 53 = 2 x 2 x 2 x 5 x 5 x 5 (WE CANNOT SIMPLIFY HERE LIKE WE DID PREVIOUSLY AS IT DOES NOT MAKE SENSE TO DO SO)
WHY DO WE NEED TO KNOW THIS?
It is sometimes more convenient to simplify things and write them in a shorter way, like explained in the indices part 1 page.
What is faster and easier to write 6 x 63 x 67 OR 611? It is 611
6 = 61 (any number to the power of 1 is just the same number) 63 = 6 x 6 x 667 = 6 x 6 x 6 x 6 x 6 x 6 x 6
This tells us that when we multiply index numbers with the same base we can add the powers of each index number. Then we can write the base to the power of the result of that addition.
e.g. If we need to simplify 52 x 54 , we can first add the powers (2 + 4 = 5) and then write the base (which is 5 in this case) to the power of the result of the addition. Thus we can write 55.
DIVIDING
DIVIDING
Question 1: Simplify 63/62
63/62 = (62 x 6)/62 = 62/62 x 6 = 1 x 6 = 6
The idea is to write the bigger number (63), which in this case is the numerator, as a product of the smaller number (62), which in this case is the denominator, and the (base)n, remember the base in the above example is 6.
So
63 = 62 x (base)n
The base in our case is 6, so we can write
63 = 62 x (6)n, clearly n = 1, we know this has we can add the powers, as learnt above.
So 63 = 62 x 6
That is how we get
63/62 = (62 x 6)/62 = 62/62 x 6 = 1 x 6 = 6
Question 2: Simplify 75/78
The bigger number is 78 and the smaller number is 75. This is obvious, because the bigger the power, the bigger the number when the base, in this case which is 7, is the same.
Now lets find a way to write the bigger number (78) as a product of the (base)n and the smaller number 75.
78 = 75 x (base)nWe know that the base is 7, so we can write
78 = 75 x 7n
How do we find n?
Given, what we have learned in the multiplication section above, adding 5 and n (which are the powers) needs to equal 8, so n equals 3.
So 78 = 75 x 73
So now we can rewrite the question as
75/78 = 75/(75 x 73) = 75/75 x 1/73 = 1 x 1/73 = 1/73 = 7-3
Being able to manipulate expressions is important in maths, we have purposely focussed on using 75 because you can see above we have been able to get (7^5)/(7^5), this gives us 1 as they cancel each other out.
ANOTHER WAY TO THINK IS DISCUSSED BELOW
Simplify 53/54
53/54 = (5 x 5 x 5) / (5 x 5 x 5 x 5) = 5/5 x 5/5 x 5/5 x 1/5
You can see that we have rewritten (5 x 5 x 5) / (5 x 5 x 5 x 5) as 5/5 x 5/5 x 5/5 x 1/5. Multiply the fractions and see that you will get exactly (5 x 5 x 5) / (5 x 5 x 5 x 5).
5/5 x 5/5 x 5/5 x 1/5 = 1 x 1 x 1 x 1/5 = 1/5 = 5-1
What we are essentially doing is cancelling out the fives; there are three fives being multiplied with each other on the numerator and four fives being multiplied with each other on the denominator.
(5 x 5 x 5) / (5 x 5 x 5 x 5) = 1/5
You can see that the fives in the purple text colour cancel each other out. This leaves us with 1/5 at the end.
Lets try another question, 534/537?
We know that there would 34 fives being multiplied by each other on the numerator, and 37 fives being multiplied by each other on the denominator, the 34 fives out the 37 fives on the denominator and the 34 fives on numerator would cancel each other out, so we are left with 3 fives being multiplied on the denominator. This shows that
534/537 = 5-3 = 1/53 = 1/125
So essentially we can subtract the powers (34-37) to get -3 as the power.
This tells us when we divide index numbers with the same base we can subtract the powers of each index number. Then we can write the base to the power of the result of that subtraction.
e.g. If we need to simplify 53/54, we can first subtract the powers (3 - 4 = -1) and then write the base (which is 5 in this case) to the power of the result of the subtraction. Thus we can write 5-1.
Indices to the power of another power
We see that the third rule in the video shows an index number to the power of n. Lets go through the two examples (83)2 and (64)5 .Indices to the power of another power
What does (83)2 mean?
(83)2 = 83 x 83 = 86 (Remember we can add the powers as explained above)
So essentially we can multiply the 3 and 2 together
Does this seem familiar? If you think about, when we write out (83)2 = 83 x 83 = 86 we are really just using the multiplication rule discussed above. In this case though, the base is an index number.
What does (64)5 mean?
(64)5 = 64 x 64 x 64 x 64 x 64
64 x 64 x 64 x 64 x 64 = 620 = 6(4 x 5)
Try these questions:
What is the value of 109/108?
What is the value of 6-2 x 66?
What is the value (4-2 x 44)/41?
Answers:
109/108 = 109 ÷ 108 = 10(9-8) = 101 = 10
6-2 x 66 = 6(-2 + 6) = 64 = 6 x 6 x 6 x 6 = 1296
Another way of expressing 6-2 x 66 is 1/62 x 66 .
6-2 x 66 = 1/62 x 66
Why? Well, remember what we went through in the indices part 1 page. 6-2 = 1/62
1/62 x 66 = 1/62 x 66/1 = 66/62 = 66 ÷ 62 = 6(6-2) = 64 = 1296
(4-2 x 44)/41 = 42/41 = 42 = 4 x 4 = 16
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