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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 1 Example 2 From the definition of $a^m = a \tim a \tim a \tim ... \tim a \\ \ \ \ \ \ \ \ \ \lefta\ \ \ m \ \ \ \ri$ The following results can be found $a^m \tim a^n = a^{m + n}$ $a^m \div a^n = a^{m-n}$ $\frac{1} {a^m} = a^{-m}$ $(a^m)^n = a^{mn}$ $\sqrt[m]{a} = a^{\frac{1} {m}}$ $\sqrt[m]{a^n} = a^{\frac{n} {m} }$ $a^0 = 1$ Example : Simplify $(2p^{ - 2} )^3 \div \left( {4p^3 } \right)^2$  $= 2^3 (p^{ - 2} )^3 \div (4^2 (p^3 )^2 ) \\ = \frac{{2^3 p^{ - 6} }}{{4^2 p^6 }} = \frac{{p^{ - 12} }}{2} = \frac{1}{{2p^{12} }}$ 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 $a^m \tim a^n = a^{m + n}$. $a^m \tim a^n = a \tim a \tim a \tim ... \tim a \ \tim \ a \tim a \tim a \tim ... \tim a \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \lefta\ \ \ m \ \ \ \ri \ \tim \lefta\ \ \ \ n \ \ \ \ri$ (by definition) So $a^m \tim a^n = a \tim a \tim a \tim a \tim a \tim .. \tim a\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \lefta\ m \ \ + \ \ n \ \ \ri$ (removing brackets) Therefore $a^m \tim a^n = a^{m + n}$ (by definition).