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## Chapter summary

Presentation: 2F44

• Exponential notation means writing a number as $${a}^{n}$$ where $$n$$ is any natural number and $$a$$ is any real number.

• $$a$$ is the base and $$n$$ is the exponent or index.

• Definition:

• $${a}^{n}=a\times a\times \cdots \times a \enspace \left(n \text{ times}\right)$$

• $${a}^{0}=1$$, if $$a\ne 0$$

• $${a}^{-n}=\dfrac{1}{{a}^{n}}$$, if $$a\ne 0$$

• $$\dfrac{1}{a^{-n}} = a^{n}$$, if $$a\ne 0$$

• The laws of exponents:

• $$a^{m} \times a^{n} = a^{m + n}$$

• $$\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}$$

• $${\left(ab\right)}^{n}={a}^{n}{b}^{n}$$

• $${\left(\dfrac{a}{b}\right)}^{n}=\dfrac{{a}^{n}}{{b}^{n}}$$

• $${\left({a}^{m}\right)}^{n}={a}^{mn}$$

• When simplifying expressions with exponents, we can reduce the bases to prime bases or factorise.
• When solving equations with exponents, we can apply the rule that if $$a^{x}=a^{y}$$ then $$x=y$$; or we can factorise the expressions.