1 Index notation

The number $4\times 4\times 4$ is written, for short, as $4^3$ and read ‘4 raised to the power 3’ or ‘4 cubed’. Note that the number of times ‘4’ occurs in the product is written as a superscript. In this context we call the superscript 3 an index or power . Similarly we could write

$5\times 5=5^2,\text{ read ‘5 to the power 2’ or ‘5 squared’}$

and

$7\times 7\times 7\times 7\times 7=7^{5}a\times a\times a=a^3,m\times m\times m\times m=m^4$

More generally, in the expression $x^y$ , $x$ is called the base and $y$ is called the index or power. The plural of index is indices . The process of raising to a power is also known as exponentiation because yet another name for a power is an exponent . When dealing with numbers your calculator is able to evaluate expressions involving powers, probably using the $x^y$ button.

Example 12

Use a calculator to evaluate $3^{12}$ .

Solution

Using the $x^y$ button on the calculator check that you obtain $3^{12}=531441$ .

Example 13

Identify the index and base in the following expressions.  

1. $8^{11}$ ,
2. ${\left(-2\right)}^5$ ,
3. $p^{-q}$

Solution

1. In the expression $8^{11}$ , 8 is the base and 11 is the index.
2. In the expression ${\left(-2\right)}^5$ , $-2$ is the base and 5 is the index.
3. In the expression $p^{-q}$ , $p$ is the base and $-q$ is the index. The interpretation of a negative index will be given in sub-section 4 which starts on page 31.

Recall from Section 1.1 that when several operations are involved we can make use of the BODMAS rule for deciding the order in which operations must be carried out. The BODMAS rule makes no mention of exponentiation. Exponentiation should be carried out immediately after any brackets have been dealt with and before multiplication and division. Consider the following examples.

Example 14

Evaluate $7\times 3^2$ .

Solution

There are two operations involved here, exponentiation and multiplication. The exponentiation should be carried out before the multiplication. So $7\times 3^2=7\times 9=63$ .

Example 15

Write out fully

1. $3m^4$ ,
2. ${\left(3m\right)}^4$ .

Solution

1. In the expression $3m^4$ the exponentiation is carried out before the multiplication by 3. So

   $3m^4\text{ means }3\times \left(m\times m\times m\times m\right)$ that is $3\times m\times m\times m\times m$

2. Here the bracketed expression is raised to the power 4 and so should be multiplied by itself four times:

   ${\left(3m\right)}^4=\left(3m\right)\times \left(3m\right)\times \left(3m\right)\times \left(3m\right)$

   Because of the associativity of multiplication we can write this as

   $3\times 3\times 3\times 3\times m\times m\times m\times m$ or simply $81m^4$ .

   Note the important distinction between ${\left(3m\right)}^4$ and $3m^4$ .

Exercises

1. Evaluate, without using a calculator,
   1. $3^3$ ,
   2. $3^5$ ,
   3. $2^5$ .
   4. $0.2^2$ ,
   5. $15^2$ .
2. Evaluate using a calculator
   1. $7^3$ ,
   2. ${\left(14\right)}^{3.2}$ .
3. Write each of the following using index notation:
   1. $7\times 7\times 7\times 7\times 7$ ,
   2. $t\times t\times t\times t$ ,
   3. $\frac{1}{2}\times \frac{1}{2}\times \frac{1}{7}\times \frac{1}{7}\times \frac{1}{7}$ .
4. Evaluate without using a calculator. Leave any fractions in fractional form.
   1. ${\left(\frac{2}{3}\right)}^2$ ,
   2. ${\left(\frac{2}{5}\right)}^3$ ,
   3. ${\left(\frac{1}{2}\right)}^2$ ,
   4. ${\left(\frac{1}{2}\right)}^3$ ,
   5. $0.1^3$ .

Answer