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Rules of Indices
1. Rules of Indices are on page 21 in the log tables.
2. Always rewrite as a number to a power e.g. 2, 3,
4
etc. etc.
3. Make sure all the base numbers are the same, and on the same level then you can let
powers equal each other.
e.g.1
e.g.2
52x – 1 = 125
52x – 1 = 53
2x – 1 = 3
2x = 4
X=2
9x
=
(32) x
27
Rewrite all to the same base
and then let power equal
each other!!
Rule
a a
1
2
3
33
=
1
32
32x
32x
=3
3
2
1
2
1
2
=3
2x = 2.5
5
X = or 1.25
4
When dividing bases
subtract the powers!!
Surds

ab  a . b
E.g.
50  25. 2

a a a
E.g.
7. 7  7

4

9
E.g.
4
9

or also 5 2
2
3

Treat surds as if they are X’s when adding or subtracting.

E.g.
also
5 2 3 2 8 2
10 2  4 2  6 2
Rewrite all to same surd so you can add or subtract

Surds are multiplied by surds and numbers by numbers.
E.g.
3 3 (2 3  3 2 )
6(3) + 9 ( 6 )
18 + 9 ( 6 )

Simplifying fractions with surds:
If there is a surd in denominator then multiply above and below by surd for it to disappear.
E.g.

9
3

9
3
*
3
3

9 3
3 3
3
To solve surds you square everything to get rid of the
3x 10  X
E.g.1
( 3 x  10 ) 2  X 2
3 x  10  X 2
X2-3x-10 = 0
(x+2)(x-5)=0
X=-2 
x=5

If you subbed back in it wouldn’t work out for -2.
E.g.2
3x  5  x  1
( 3x  5 ) 2  ( x  1) 2
3x  5  ( x  1)( x  1)
3x  5  x  2 x  1
2
X2 – 5x + 6 = 0
(x – 2) (x – 3) = 0
X=2 X=3
Scientific Notation
This involves writing big numbers in index form or scientific notation.
E.g.
56,000,000,000,000
is also
5.6
*
1013
Number between
1 and 10
Written as a
power of 10
e.g.
260,000 = 2.6 * 105 (move decimal point 5 places to the left)
0.00372 = 3.72 * 10-3 (move decimal place to the right by 3)
1.
Addition and Subtraction

Write each number as a natural number (positive whole numbers)

Add or subtract the numbers

Write them in index notation / scientific form
E.g.1
5.328 * 105 minus 2.8 * 103
532,800
2,800
530,000
5.3 * 105
E.g.2 6.28 * 10-2
plus
0.0628
plus
0.066
Answer is 6.6*10-2
3.2 * 10-3
0.0032
2. Multiplication and Division

Multiply or divide the numbers

Multiply or divide the powers of 10 (add or subtract the indices)

Write them in index notation / scientific form
E.g.1 (6.5 * 103) multiplied by (2.4 * 102)
6.5 * 2.4
= 15.6
15.6 * 105
E.g.2 (3.91*104)
(1.7 * 10-1)
3.91
 2 .3
1 .7
2.3 * 105
10 4
 10 41  10 5
10 1
103 * 102
= 105