Exponents Introduction

Using an exponent is a short hand form of writing repeated multiplication or repeated division.

The idea of short hand forms is not new: multiplication is a short hand form of writing repeated addition. For example:

    \[2+2+2+2+2+2+2+2=8\times2=16\]

Writing 8\times2 is much easier than writing out the long addition.

As with any new notation in mathematics, we need to know what it means before we can use it well.

Here is an example of repeated multiplication:

    \[2\times2\times2\times2\times2\times2\times2\times2=256\]

There are eight twos on the left hand side. This is how we use exponent notation for this repeated multiplication:

    \[2^8=256\]

The other thing we know about multiplication is that we can multiply numbers in any order (commutativity property of real numbers).

    \[2\times3=3\times 2=6\]

Now let’s use the commutativity property, and our new exponent notation to simplify this long expression:

    \begin{align*}&2 \times 3 \times 4\times 2\times 4\times4 \times 2\times 4\times 4 \\=&2\times2\times2\times3\times4\times4\times4\times4\times4 \\=&2^3\times3\times4^5 \end{align*}

Video from Math Antics

A Sequence: Repeated Multiplication; Repeated Division

Move the slider to the right to multiply by 2.

Move the slider left to divide by 2.

Notice the ‘middle number’ is 1.

applet link

An exponent records the number of times that the number 1 is multiplied or divided by the same number.

    \begin{align*}5^4&=1\times 5\times 5\times 5 \times 5 &= 625\\[10pt]5^3&=1\times 5\times 5\times 5&=125 \\[10pt]5^2&=1\times 5\times 5&=25\\[10pt]5^1&=1\times 5&=5\\[10pt]5^0&=1&=1\\[10pt]5^{-1}&=1\div 5&=\frac{1}{5} \\[10pt]5^{-2}&=1\div 5 \div 5 &=\frac{1}{25}\\[10pt]5^{-3}&=1\div 5\div 5\div 5&=\frac{1}{125} \end{align}

You can type in any exponent of 5 into the calculator and arrive at any other exponent of 5 by either repeated multiplication by 5 or repeated division by 5. To multiply is to add to the exponent. To divide is to take from the exponent.

To learn about exponents, its useful to have some exponent facts memorized. The familiar values are square numbers, cube numbers, the first ten powers of 2 and the first 5 powers of 3.

Complete the table on Exponent Number Facts to Memorize to see how many exponent number facts you already have. The solution pdf is ENFM Solutions. Then test yourself with the following applet.
applet link

Notation

There are three words used for an exponent, they are used interchangeably. These are: exponent, power and index.

Introduction to exponents: vocabulary

Why Exponents?

By learning about exponents we lay the foundation for several areas of learning. Here are four:

Math Description
Scientific Notation A way of measuring exceedingly small values (the weight of a fruit fly) and exceedingly large values (the speed of light) using familiar units (kg, meters etc).
Growth and Decay Exponential functions allow us to mathematically model how populations (eg of people, of bats, of bacteria) grow; and to model how materials like uranium decay over time.
Money Savings accounts and loans, left to themselves, increase exponentially. To understand how this works, we need to understand exponents.
Prime Factors Finding prime factors is a little like finding out the ingredients in a good soup. Prime factors help us see how numbers are related to each other.

For example, 24=2^3\times 3; and the number 40=2^3 \times 5.

Number Sense, compare powers of 2 and 4:

Compare the first five powers of 4 with the first ten powers of 2. Explain the pattern.  Choose various values of m and n such that the following equation is true:

    \[2^m=4^n\]

Number Sense, compare powers of 3 and 9:

Write three different equations, choosing values for m and n carefully such that the equation

    \[3^m=9^n\]

is true. Check your equations with a calculator.

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