Sectieoverzicht

  • What are exponents?

    Exponents are a powerful way to mathematically describe rapid increases or decreases in growth. Exponential notation is very useful to describe very large and very small numbers.

    Just as multiplication is a short way to write repeated addition, similarly, exponents are a short way to write repeated multiplication. We discuss this further in the example that follows the video. 

    The power of exponential growth can be seen using a chessboard and grains of rice watch the next video to see this in action.

    • Leerlingen moeten
      Als voltooid aanduiden

      Example

      Look at the following expressions. Which do you think would be easier to write and work with? \(2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\) or \(2^{10}\)?
      Certainly \({{2}^{10}}\) is quicker to write and leaves less room for error.

      But how did we get from \(2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\) to
      \(2^{10}\)?

      Count the number of times that \(2\) has been multiplied by itself. What do you get?

      You would have counted that \(2\) is multiplied \(10\) times. The number ‘floating above’ the \(2\), which is called the exponent, is also \(10\). The exponent tells us how many times the base \(2\) has been multiplied by itself.

      \(2^{10}\) is called exponential form or a power. The base is the number that is repeatedly multiplied. The exponent which may also be referred to as the ‘index’ or ‘power’ tells us how many times the base is multiplied by itself.

      Exponential form explained

      We read \(2^{10}\) as \(2\) to the power of \(10\). This tells us that the base \(2\) has been multiplied by itself \(10\) times

      Think you've got it? Then try the activity on writing in exponential form.

    • Activity: Write in exponential form Test
      Leerlingen moeten
      Als voltooid aanduiden
    • Leerlingen moeten
      Als voltooid aanduiden

      Exponential identities

      There is an important identity that we use often in exponents. The identity states that any base raised to the power of zero is equal to one or \({{a}^{0}}=1\). 

      So far in the examples you have mainly used positive exponents but exponents can be negative too. Negative exponents produce fractions. You will use the following identity when dealing with negative exponents: \({a}^{-m}=\frac{1}{{a}^{+m}}\).

      The table below will help you understand how to change from negative to positive exponents.

      Converting negative exponents
      NEGATIVE EXPONENT    AS POSITIVE EXPONENT       SIMPLIFIED
      \({{3}^{-2}}\) \({{3}^{-2}}=\frac{1}{{{3}^{+2}}}\) \({{3}^{-2}}=\frac{1}{{{3}^{+2}}}=\frac{1}{9}\)
      \({{(2x)}^{-2}}\) \({{(2x)}^{-2}}=\frac{1}{{{(2x)}^{2}}}\) \({{(2x)}^{-2}}=\frac{1}{{{(2x)}^{2}}}=\frac{1}{4{{x}^{2}}}\)
      \(\frac{1}{{{2}^{-3}}}\) \(\frac{1}{{{2}^{-3}}}={{2}^{+3}}\) \(\frac{1}{{{2}^{-3}}}={{2}^{+3}}=8\)


      Click on the link below to take you to the next activity to check your understanding of negative exponents.

      Negative exponents practise