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.