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Why are there letters in maths? Well, to represent numbers that we do not know, we can represent them by letters. For example, the result in your next examination is an unknown number so we can call your future result x.
So as in our example in video 1, we have the expression written below.
a + a + 2c + b + a + c
Lets explain what this means. The letter a represents an unknown number, the letter b represents an unknown number and the letter c represents an unknown number. WHETHER EACH LETTER REPRESENTS THE SAME NUMBER AS EACH OTHER OR COMPLETELY DIFFERENT NUMBERS, WE DO NOT KNOW, HOWEVER WE CAN SIMPLIFY THE EXPRESSION.
a + a + a = 3a = 3 x a 2c + c = 3c = 3 x c b = b
Collecting like terms just means collecting the same unknown numbers represented by the letters.Think about it logically. If somebody gave you three £5 notes. Would you say the person gave you £5 + £5 + £5 or would you say the person gave you £15. £15, right?
In maths, writing a + a + a or 3a is the same thing as they EQUAL each other, however, writing 3a takes less space and simplifies the expression.
Why does 2c + c = 3c?
Think about it logically, the letter c represents an unknown number, so if we have 2 of that number plus the same unknown number again, we will have 3 of that unknown number.
Also, remember c + c = 2c
There is only 1 b.
Soa + a + 2c + b + a + c = 3a + 3c + b
Question 1: a + a =
Answer is 2a
a represents an unknown number, so by adding two of the same unknown number we get 2 multiplied that unknown number.
Question 2: ab + ab = 2ab
Answer is 2ab
When we write ab, a and b represent unknown numbers, what we are actually saying is
ab = a x b
ab represents a number because a x b gives you a product. So having two of that product means you would have 2 times that product. We can add the ab's together because ab is a number itself (it is the product of a multiplied by b).
ab + ab = 2ab = 2 x a x b
Lets say we had another question which we needed to simplify; this is written below.
Simplify the expression ab + ab + abc
We know that ab = a x b (so ab is a product of the unknown numbers a and b) abc = a x b x c (so abc is a product of the unknown numbers a, b and c)
ab and abc are two different numbers (so how can we simplify the expression? Well, we CAN'T)
So ab + ab + abc = 2ab + abc
Question 3: a2 + a =
Answer is a2 + a
In this question there was nothing to simplify. Why?
Well,
a represents an unknown number (you know this)
a2 represents the unknown number a squared.
You can see that that a and a2 are two different numbers, so we cannot simplify the expression, like in the example of question 2, ab and abc are two different numbers so we cannot simplify them.
Lets try another question.
Simplify the expression a3 + a4 - a3
Lets see which each term means
a3 = a x a x a a4 = a x a x a x a
You can see that a3 and a4 are different unknown numbers. This is obvious too, because the base a is put to different powers.
Lets now simplify and collect like terms.
a3 + a4 - a3 = a4
Why?
a3 - a3 = 0
So we are left with only a4 .
If you have a number and then take away the same number you will get 0) e.g. if you have 5 and then takeaway 5 you would get 0.
a4 = a4 (there are no other like terms to collect, as there is only 1 a4)
THE BEST WAY TO SIMPLIFY AND NOT MAKE MISTAKES IS BY ACTUALLY SEEING WHAT EACH TERM ACTUALLY MEANS.
Question 4: ab2 + a2b =
Answer is ab2 + a2b
ab2 = a x b2 = a x b x b (Remember b2 = b x b)
a2b = a2 x b = a x a x b (Remember a2 = a x a)
So now we know what each term means, we can see that they are different, we cannot simplify as a result.
so ab2 + a2b = ab2 + a2b
Lets say we have another question which we needed to simplify.
Simplify a3b + a2b2 - 3ba3
Lets see what the terms mean:
a3b = a3 x b = a x a x a x b (Remember a3 = a x a x a)
a2b2 = a2 x b2 = a x a x b x b
-3ba3 = -3 x a3 x b = -3 x b x a x a x a
So we can simplify the expression
a3b + a2b2 -3ba3 = -2a3b + a2b2 (ANSWER)
Why?
a3b -3ba3 = -2a3b
a2b2 = a2b2 (no other like terms, so we cannot simplify)
You might be thinking how a3b -3ba3 = -2a3b.
a3b = a3 x b = a x a x a x b
-3ba3 = -3 x a3 x b = -3 x b x a x a x a
But YOU NEED TO NOTICE THAT -3 x a3 x b = -3 x b x a3
(WHEN WE MULTIPLY WE CAN SWITCH AROUND WHAT COMES FIRST IN THE PRODUCT. E.G. 5 X 4 = 4 X 5)
so
a3b -3ba3 = a3b -3a3b = -2a3b
Question 5: cd + cde =
Answer is cd + cde
cd = c x dcde = c x d x e
So cd and cde are two different numbers (look at what they mean above) so we cannot simplify.
Question: 2a + a + c2 + a2 =
Answer is 3a + c2 + a2
Lets look at the terms
2a + a = 3a (Remember 2a = a + a )
c2 = c x c (there are no other terms with c2, so we cannot simplify any further)
a2 = a x a (there are no other terms with a2, so we cannot simplify any further)
Lets say we have another expression to simplify; this is written below.
2ab - 4abc - 4ba
2ab = 2 x a x b
-4abc = -4 x a x b x c
-4ba = -4 x b x a = -4 x a x b = -4ab
So we can simplify the ab term
2ab - 4ba = 2ab -4ab = -2ab
-4abc = -4abc (there is no need to simplify)
so 2ab - 4abc - 4ab = -2ab - 4abc
Video 1 Explanation
Video 1 Explanation
So as in our example in video 1, we have the expression written below.
a + a + 2c + b + a + c
Lets explain what this means. The letter a represents an unknown number, the letter b represents an unknown number and the letter c represents an unknown number. WHETHER EACH LETTER REPRESENTS THE SAME NUMBER AS EACH OTHER OR COMPLETELY DIFFERENT NUMBERS, WE DO NOT KNOW, HOWEVER WE CAN SIMPLIFY THE EXPRESSION.
a + a + a = 3a = 3 x a 2c + c = 3c = 3 x c b = b
Collecting like terms just means collecting the same unknown numbers represented by the letters.Think about it logically. If somebody gave you three £5 notes. Would you say the person gave you £5 + £5 + £5 or would you say the person gave you £15. £15, right?
In maths, writing a + a + a or 3a is the same thing as they EQUAL each other, however, writing 3a takes less space and simplifies the expression.
Why does 2c + c = 3c?
Think about it logically, the letter c represents an unknown number, so if we have 2 of that number plus the same unknown number again, we will have 3 of that unknown number.
Also, remember c + c = 2c
There is only 1 b.
Soa + a + 2c + b + a + c = 3a + 3c + b
Video 2 EXPLANATION
Video 2 EXPLANATION
Question 1: a + a =
Answer is 2a
a represents an unknown number, so by adding two of the same unknown number we get 2 multiplied that unknown number.
Question 2: ab + ab = 2ab
Answer is 2ab
When we write ab, a and b represent unknown numbers, what we are actually saying is
ab = a x b
ab represents a number because a x b gives you a product. So having two of that product means you would have 2 times that product. We can add the ab's together because ab is a number itself (it is the product of a multiplied by b).
ab + ab = 2ab = 2 x a x b
Lets say we had another question which we needed to simplify; this is written below.
Simplify the expression ab + ab + abc
We know that ab = a x b (so ab is a product of the unknown numbers a and b) abc = a x b x c (so abc is a product of the unknown numbers a, b and c)
ab and abc are two different numbers (so how can we simplify the expression? Well, we CAN'T)
So ab + ab + abc = 2ab + abc
Question 3: a2 + a =
Answer is a2 + a
In this question there was nothing to simplify. Why?
Well,
a represents an unknown number (you know this)
a2 represents the unknown number a squared.
You can see that that a and a2 are two different numbers, so we cannot simplify the expression, like in the example of question 2, ab and abc are two different numbers so we cannot simplify them.
Lets try another question.
Simplify the expression a3 + a4 - a3
Lets see which each term means
a3 = a x a x a a4 = a x a x a x a
You can see that a3 and a4 are different unknown numbers. This is obvious too, because the base a is put to different powers.
Lets now simplify and collect like terms.
a3 + a4 - a3 = a4
Why?
a3 - a3 = 0
So we are left with only a4 .
If you have a number and then take away the same number you will get 0) e.g. if you have 5 and then takeaway 5 you would get 0.
a4 = a4 (there are no other like terms to collect, as there is only 1 a4)
THE BEST WAY TO SIMPLIFY AND NOT MAKE MISTAKES IS BY ACTUALLY SEEING WHAT EACH TERM ACTUALLY MEANS.
Question 4: ab2 + a2b =
Answer is ab2 + a2b
ab2 = a x b2 = a x b x b (Remember b2 = b x b)
a2b = a2 x b = a x a x b (Remember a2 = a x a)
So now we know what each term means, we can see that they are different, we cannot simplify as a result.
so ab2 + a2b = ab2 + a2b
Lets say we have another question which we needed to simplify.
Simplify a3b + a2b2 - 3ba3
Lets see what the terms mean:
a3b = a3 x b = a x a x a x b (Remember a3 = a x a x a)
a2b2 = a2 x b2 = a x a x b x b
-3ba3 = -3 x a3 x b = -3 x b x a x a x a
So we can simplify the expression
a3b + a2b2 -3ba3 = -2a3b + a2b2 (ANSWER)
Why?
a3b -3ba3 = -2a3b
a2b2 = a2b2 (no other like terms, so we cannot simplify)
You might be thinking how a3b -3ba3 = -2a3b.
a3b = a3 x b = a x a x a x b
-3ba3 = -3 x a3 x b = -3 x b x a x a x a
But YOU NEED TO NOTICE THAT -3 x a3 x b = -3 x b x a3
(WHEN WE MULTIPLY WE CAN SWITCH AROUND WHAT COMES FIRST IN THE PRODUCT. E.G. 5 X 4 = 4 X 5)
so
a3b -3ba3 = a3b -3a3b = -2a3b
Question 5: cd + cde =
Answer is cd + cde
cd = c x dcde = c x d x e
So cd and cde are two different numbers (look at what they mean above) so we cannot simplify.
Video 3 EXPLANATION
Question: 2a + a + c2 + a2 =Video 3 EXPLANATION
Answer is 3a + c2 + a2
Lets look at the terms
2a + a = 3a (Remember 2a = a + a )
c2 = c x c (there are no other terms with c2, so we cannot simplify any further)
a2 = a x a (there are no other terms with a2, so we cannot simplify any further)
Lets say we have another expression to simplify; this is written below.
2ab - 4abc - 4ba
2ab = 2 x a x b
-4abc = -4 x a x b x c
-4ba = -4 x b x a = -4 x a x b = -4ab
So we can simplify the ab term
2ab - 4ba = 2ab -4ab = -2ab
-4abc = -4abc (there is no need to simplify)
so 2ab - 4abc - 4ab = -2ab - 4abc
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