Algebra

1. You know that a³ is a short hand way of writing a × a × a.
   Write the following expressions out fully like the example above.

   a) a5	b) a3b2	c) a5 b3	d) a4 × a2	e)a3 × a6
   f) a3 × a2	g) a5 × a2	h) b4 × b3	i) (a3)2	j) (a2)4

2. In questions 1 d, e, f, g and h, you will have noticed that they were expressed in the form a^m × aⁿ or b^m × bⁿ and therefore they were not written in the shortest form. For these questions, write them out in their shortest form.

   What did you notice? How can you write a^m × aⁿ in their shortest form?

3. By first writing the expressions out in full then cancelling, simplify these expressions. The first one is done for you.

   a) a5 ÷ a2 = a × a × a × a × a ÷ (a × a) = a × a × a × a× a÷ (a × a) = a × a × a = a3	b) b3 ÷ b2	c) b5 ÷ b3
   d) b7 ÷ b3	e) b8 ÷ b3

   What do you notice about b^m × bⁿ? How can you write them in their shortest form?

   You should by now have worked out that

   a. a^m × aⁿ = a^m+n
   b. a^m ÷ aⁿ = a^m-n

   Substitute a,m and n with numbers of your choice and fiddle about with them to convince yourself.

   Using these results this implies that
   

   c. a^m ÷ a^m = a⁰
   d. So a⁰=1

   Combining results b,c and d gives us
   

   e. 1÷ a^m = a⁰ ÷ a^m
   f. = a^-m

   What do you think fractional indices mean then? I.e. if you had a^1/2 or a^1/3 etc.

   This might help you:

4. a) a^1/2 × a^1/2 = ...
   b) a^1/3 × a^1/3 × a^1/3=

Check your answers here.
We'll give indices a break now while you continue with the other bits.

Before you try the next exercise, I would advise you to read up on the algebra notes on simplification first.

5. Simplify the following expressions (collect the like terms).

   a) 5a + 3b - 4a + 2b + c	b) 6a2 + 2ab - a + 2a2
   c) 6b2 + 3a2 + 2a2 + 2ab + 4ab	d) 10a + 6b + 2a + 3a2 + 3b
   e) ab2 - b2 + 4ab2

6. a) Write out an algebraic expression for the length of the perimeter of the trapezium shown.
   b) Write out an expression for its area by considering the area of the shapes within the trapezium.
   
   
   
7. A magic square is one whose total at the end of each row, column and diagonal adds up to the same number.
   Show that the square below is a magic square. Remember to simplify your answer.

   m+p	m-p+q	m-q
   m-p-q	m	m+p+q
   m+q	m+p-q	m-p

8. Expand the following brackets.

   a) 4(a + 3b)	b) a2(3a + ab)	c) 2ab(ab + b2)
   d) 3a3(4a + 3x)	e) ab2(5a + b3)

   Back to algebra

   Top

   Home