What are Indices?
In Mathematics, Indices are simply “repeated multiplication”. There are 6 Index laws which needs to be remembered. A basic example showing each part of the equation is shown below:
The Index Laws
First Index Law:
am × an = am + n
Example:
a5 × a3 = a8
Second Index Law:
am / an = am – n
Example:
a5 / a3 = a2
Third Index Law:
a0 = 1 (where a ≠ 0)
Example:
Step 1: Simplify the following expression:
43 / 43
Step 2: Use the second law of indices:
43 / 43 = 4 3-3 = 40
Step 3: Expand the numerator and denominator:
43/43 = (4 × 4 × 4) / (4 × 4 × 4)
Step 4: We cancel the “4” and are left with the following:
(1 × 1 × 1) / (1 × 1 × 1) = 1
Step 5: Therefore 40 = 1
Fourth Index Law:
(am) = am × n
Example:
(a5)3 = a15
Fifth Index Law:
(a × b)m = am × bm
Example:
(5 × 3)5 = 55 × 35
Sixth Index Law:
(a / b)m = am / bm
Example:
(5 / 6)3 = (5/6) × (5/6) × (5/6)
= (5 × 5 × 5) / (6 × 6 × 6)
= 53 / 63
Negative Indices:
a-n = 1 / an (where a≠0)
Example:
1 / 63 = 60/63
= 60-3
= 6-3
Roots
Roots are the opposite of exponents that we have been looking at above. You can reverse an exponent with a “root”. For example, if you square 3 you will get 9. If you then square root 9 you will get 3.
√a = a1/2
Example:√9 = 3
3√a = a 1/3
Example:3√64 = 4
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