Chapter summary
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2.4 Exponential equations

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Chapter summary

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:

\(a^{m} \times a^{n} = a^{m + n}\)

\(\dfrac{{a}^{m}}{{a}^{n}}={a}^{mn}\)

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

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

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

 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.
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2.4 Exponential equations

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