Mathematics » Exponents and Surds » Exponential Equations

Summary and Main Ideas

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:

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

    • \({a}^{0}=1\), 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:

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

    • \(\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}\)

    • \({(ab)}^{n}={a}^{n}{b}^{n}\)

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

    • \({({a}^{m})}^{n}={a}^{mn}\)

  • 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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