1. Incredible Indices

2. 8x8 ADD SUBTRACT When we multiply, we the powers When we divide, we
Note the c terms cancel out completely because they have the same powers!

3. Remember to place the letters in the top or bottom depending on where the biggest power was !
Remember after you do all cancelling, and there’s nothing left in the TOP, you MUST put a 1.
Remember after you do all cancelling, and there’s nothing left in the BOTTOM, you DON’T have to put a 1.

4. 1 Anything raised to power zero equals… 50 = b0 + 7 = 8 1 a0 = 12
( – 356)0 = 1
4a0 =
4  x0
(2ab2c8)0 = 1
= 4  1
= 4
a0 + 3b0 + 5c0 = 9

5. MULTIPLY means which is The quick way to get the answer is to
Raising powers to powers using brackets
means
which is
The quick way to get the answer is to
MULTIPLY
the powers!!

6. Combined operations 1 3 = = = =

7. Combined operations 2 3
Remember with divisions you need to turn the 2nd fraction upside down!
This is called multiplying by the reciprocal. 8 = =

8. Negative Powers a – 3 = b – 7 = (3b) – 2 = 3b – 2 = 3  b – 2 = 3 
Note – negative power terms go to the bottom. Others to the top!!
a – 3b 4 c – 7 d

9. When a fraction is raised to a negative power, you can invert the fraction and change the power to a positive!

10. inverting the stuff in the brackets and changing the outside power from – 3 to 3
getting rid of the brackets. 53 is 125, and 23 is 8. Powers are all multiplied by 3
Moving anything with a negative power to the opposite part of the fraction.

11. = 24x – 12 (ans) More “exotic” power problems! 1.
Recognising all the big numbers as powers of 2
Removing the brackets. Remember when you remove brackets, you multiply powers
Multiplying along top & bottom. Remember when you multiply, you add powers
Now we’ll divide the top by the bottom and this means we subtract the powers. (2x – 6) – (6 – 2x) = 4x – 12
= 24x – 12 (ans)

12. = 24 × 32 2. Simplify Express your answer as powers of 2 and 3
Breaking all 4 big numbers down into their prime factors
Removing the brackets. Remember when you remove brackets, you multiply powers
Selectively multiplying terms with the same big numbers & adding powers
Now divide the top by the bottom and this means we subtract the powers (top minus bottom)
2’s: (-x + 3) – ( – 1 – x) = 4
3’s: (x – 3) – ( x – 5 ) = 2
= 24 × 32
Or…

13. Fractional powers

14. OR
Example
Using left formula OR
Using right formula
You can use either of these formulas, but the one on the right usually avoids getting really big numbers and so makes the whole process more manageable

15. Work out 82/3 using the graphics

16. Examples using fractional powers=
Note that 91/2 = 9 which is 3.
The rest of the powers have been multiplied
6 x ½ = 3 and 10 x ½ = 5

17. First, get rid of any brackets and switch terms with negative powers to the other part of the fraction
Note the y – 8 has been switched from top to bottom