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Algebra and Functions : Key Stage 5 (AS-Level) : Mathematics Content Knowledge


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Key Stage 5 (AS-Level)
Algebra and Functions
Question 1 of 19

1. How confident are you that you know and can use:

a. the laws of indices for all rational exponents?


Example



Example 1 Example 2
   

From the definition of a^m = a 	im a 	im a 	im ... 	im a\\ \ \ \ \ \ \ \ \ \lefta\ \ \ m \ \ \ 
i

The following results can be found

  • a^m 	im a^n = a^{m + n}
  • a^m \div a^n = a^{m-n}
  • \frac{1} {a^m} = a^{-m}
  • (a^m)^n = a^{mn}
  • \sqrt[m]{a}  = a^{\frac{1} {m}}
  • \sqrt[m]{a^n}  = a^{\frac{n} {m} }
  • a^0 = 1

Example : Simplify (2p^{ - 2} )^3  \div  \left( {4p^3 } 
ight)^2

 = 2^3 (p^{ - 2} )^3  \div (4^2 (p^3 )^2 ) \\
            = \frac{{2^3 p^{ - 6} }}{{4^2 p^6 }} = \frac{{p^{ - 12} }}{2} = \frac{1}{{2p^{12} }}

The laws of indices provide a good opportunity to introduce students to the notion of proof and the formal language and clear layout used by mathematicians to make a convincing argument. Here is an example:

Show that a^m 	im a^n = a^{m + n}.

a^m 	im a^n = a 	im a 	im a 	im ... 	im a \ 	im \ a 	im a 	im a 	im ... 	im a\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \lefta\ \ \ m \ \ \ 
i \ 	im  \lefta\ \ \ \ n \ \ \ 
i (by definition)

So a^m 	im a^n = a 	im a 	im a 	im a 	im a 	im .. 	im a\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \lefta\ m \ \ + \ \ n \ \ 
i (removing brackets)

Therefore a^m 	im a^n = a^{m + n} (by definition).

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