Mathematics & Statistic Tutor Perth - SPSS Help   

 Set One

Question 2

70xy + 5xy - 37gh + 45gh 

The letters are in order and all the terms are together, so we don’t have to worry about that bit.  although I do like to put all the positive (add) numbers before the negative (minus).

It then looks like:

       70xy + 5xy + 45gh - 37gh  

Remember add or subtract the numbers and then just write the letters down.

       70 + 5 is 75  and  45 - 37 = 8

So write the correct letters behind the numbers:

The answer is  75xy +8gh

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Question3

       3a - 7a + 20a - 6a                    

We only have one letter so we will order the letters with the positive(add) at the beginning.

3a + 20a - 6a - 7a

Now we can add and subtract the numbers, there are many ways of doing this but I will stick to the very simplest one.

3 + 20 = 23     then

23 -   6 = 17  then

17 -  7 = 10   finally no more

Now write the letter behind the number.

So the answer is   10 a

We could have said that we had to take away 13 (6+7) from 23 (20+3) and hence

23 - 13 =10

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Set Two

Question 2

Count the number of  w’s, and you will find there are seven

 w  x       w x       w x      w x       w x       w x       w =   w ⁷

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Question 3

b  x        b  x        b  x        b  x      b x      b x      h  x      h x      h x     h x       h

Count the number of b’s and the number of h’s.
There are 6 b’s and there are 5 h’s.

b x        b  x        b x       b  x      b x      b x      h  x      h x      h x     h x       h= b ⁶ x       h ⁵= b ⁶ h ⁵

So the answer is b⁶ h⁵

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Set Three

Question One           5n ⁴y         x           2any

Step One            

Collect the numbers together and collect all the same letters together next.

5 x      2  x      a x      n ⁴x      n x      y x      y

Step Two

Now multiply the numbers and same letters together,

       5 x     2 x       a x      n x     n x      n x      n x     n x      y x      y
 

You do not have to write all the letters out in full as you can add the number of times the letter appears,
For example        n ⁴  x         n

n ⁴  is four n’s multiplied together and n is just one n, so that makes 5 n’s multiplied together overall.  This is written as  n ⁵ .

You may notice now that when we multiply letters, we add the powers of the letters.
So lets look at that     n ⁴  x        n ¹   =    n ⁴⁺¹   =  n ⁵
The powers add together to give 5 , so we get  n ⁵

Back to the question whichever way you have approached it:

=   5 x      2 x      a x     n ⁴x     n x    y x    y

=  5 x     2 x       a x       n x       n x        n x        n x       n x       y x      y
 

=  10 x     a x      n ⁵ x     y ²  =  10an ⁵ y ² 

Step Three

Now  write numbers and letters without the multiplication signs and put in alphabetical order.

• 10 x      a x     n ⁵ x     y ²    =  10an ⁵ y ² 

The answer is 10an ⁵ y ²

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Question Two         2a ⁷ by         x        a ⁴ b ⁴y

Step One            

Collect the numbers together and collect all the same letters together next.

       2  x     a⁷ x      a ⁴   x      b x      b ⁴  x      y  x      y

Step Two

Now multiply the numbers and same letters together,

2 x    a x     a x     a x     a x     a x     a x     a x    a x    a x    a x    a x     b x     b x     b x     b x     b x     y x     y

=  2 x     a ¹¹x     b ⁵ x    y ²  

Adding the powers together we get the same result

=  2    x         a ^{(7 + 4)}   x       b ^{(1 +4)}   x        y ^{(1 +1)}
=  2   x         a ¹¹   x      b ⁵    x       y ²

You can only add the powers of the same letter to get the correct answer.

Step Three

Now  write numbers and letters without the multiplication signs and put in alphabetical order.

              2   x         a ¹¹   x       b ⁵    x       y ² =   2a ¹¹b ⁵y ²

The answer is 2a ¹¹b ⁵y ²

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Question Three         2p ¹¹x    ⁵     x          p ⁶x    ³

Step One            

Collect the numbers together and collect all the same letters together next.

       2 x    p ¹¹ x    p ⁶ x      x    ⁵ x     x    ³   

Step Two

Now multiply the numbers and same letters together,

              2 x    p x    p x    p x     p x    p x    p x    p x     p x    p x    p x    p x     p x    p x   p x    p x     p x    p x    x     x    x    x     x    x    x    x     x    x    x    x     x   

=  2 x     p ¹⁷ x     x    ⁸

This can get very  long to write out when you have big numbers for the powers. So when you are confident try to use adding the powers only.  Remember you can always go back to writing the letters out in full to check your answer.

Adding the powers together we get the same result

=  2 x    p ^{(11 + 6)} x     x    ^{(5 +3)}
=  2 x    p ¹⁷ x     x    ⁸

Step Three

Now  write numbers and letters without the multiplication signs and put in alphabetical order.

       2 x     p ¹⁷ x      x    ⁸     =   2p ¹⁷x    ⁸

The answer is 2p ¹⁷x    ⁸

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Set Four

Question One             13 b ⁴ f g ²
                                            2 b ³ f g ²

Step One           Write out  in full

13 x    b x    b x    b x    b x      f x      g x     g 
 2  x    b x     b x    b x             f x      g x      g  

Step Two         Cancel like terms

13 x    1 x    1 x     1 x      b x      1 x     1 x    1 
 2 x         x       x         x          x          x          x   

 

Step Three      Write without the multiplication signs and number ones.

13 b  
 2    

                      The answer is 13 b  
                                               2   

You do not have to write all the letters out in full as you can simply do a subtraction. The top power of the letter take away the bottom power of the same letter
For example        b ⁴    /   b ³  or it can be written    b ⁴
                                                                                            b ³

b⁴  is four b’s multiplied together and b³ is three b’s multiplied together, so we have 4 b’s on the top and 3 b’s on the bottom which will cancel to leave one b on top.

You may notice now that when we divide letters, we subtract the powers of the letters always the top take away the bottom.
So lets look at that     b ⁴   /   b ³   =    b ^{4 -  3}   =  b ¹
The powers subtract to give 1 , so we get  b.

Without too much detail, the following is usually true when subtracting powers when dividing:

Positive powers the letter remains on the top.

Negative powers the letter goes on the bottom.

You will notice that negative powers can be written as positive powers by moving the letters from the top to the bottom or from the bottom to the top.

For example  b ^{- 3} is the same as 1/ b ³  or   1
                                                                     b ³

                       1 /  b ^{- 4} is the same as 1 / (1/ b ⁴) which is the same as b ⁴

I know that last one does not look quite correct but it is believe me.

Looking at this example and using the subtraction of powers method.

                                   13 b ⁴f g ²
                                                 2 b ³f g ²

                                 13 b ^{(4-  3)} f ^{(1-  1)} g  ^{(2-  2)}
                                         2

                                 13 b ⁽¹⁾f ⁽⁰⁾g  ⁽⁰⁾    =  13b
                                         2                              2

Now we have another wierd situation with the number zero where anything raised to the power zero is one. This should not come as a big surprise as b/b is one.

      b ¹   /   b ¹   =    b ^{1 -  1}   =  b ⁰
The powers subtract to give 0 , so we get  1.

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Question Two               a ⁴ b ⁴
                                                  a ⁴ b ⁴
 

Step One           Write out  in full

a x    a x    a x    a x    b x    b x    b x    b    
a x     a x     a x     a x    b x     b x     b x     b   

Step Two         Cancel like terms

   1 x    1 x     1 x     1 x     1 x    1 x    1 x    1   
         x            x            x              x            x         x             x         

 

Step Three     Write without the multiplication signs and number ones.

The answer is 1

We have to write this one as this is the answer

 

Looking at this example and using the subtraction of powers method.

                                     a ⁴ b ⁴
                                                   a ⁴b ⁴

I like to think of the division line as the minus sign of the subtraction, remember it has to be the same letter!

                                     a ⁴b ⁴  =  a ^(4-4) b ^(4-4) =  a ⁰ b ⁰  =  1  
                                                   a ⁴b ⁴

The answer is 1

 

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Question Three                    - 20m ⁸
                                     15m ²

Step One           Write out  in full

- 20   x    m x    m x    m x    m x    m x    m x    m x    m   
  15    x     m x     m

Step Two         Cancel like terms, watch out for the minus sign. Notice we have rewritten 20 as 5 times 4 and 15 as 5 times 3 so we can cancel the fives.

-  5 x     4 x    m x    m x    m x    m x    m x    m x    m x    m   
   5 x        3  x     m x     m

-  1 x    4 x    1 x    1 x    m x    m x    m x    m x    m x    m   
      x     3 x        x         

Step Three       Write without the multiplication signs and number ones.

-  4  m ⁶
   3      

The answer is -  4  m ⁶
                         3    

Looking at this example and using the subtraction of powers method

                               - 20 m ⁸
                                 15 m ²

The number part will remain the same regardless what method you use.

                               -  4 m ^{(8 -  2)} = -  4  m ⁶
                                  3                    3     

The answer is -  4  m⁶
                         3                            

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