Index notation with variable bases

Let's start by reviewing the key concepts introduced in Exponential Notation:

• In the expression $a^n$an, $a$a is the base and $n$n is the exponent (or power)
• The exponent (or power) indicates how many times the base number is a factor

 

Exploration

Think about the equation $7^3=7\times7\times7$73=7×7×7. On the left hand side we can recognise that $7^3$73 is the exponential notation, where the base is the number $7$7 and the number $3$3 is the power.

On the right hand side we can see that $7\times7\times7$7×7×7 is the expanded notation, where the product consists of three $7$7s multiplied together. Since this is an equation, we know that the exponential notation is equivalent to the expanded notation.

What if we were not so interested in the base number $7$7 specifically? What if we wanted to talk about the general rule that is true for any base?

 

An algebraic base

Up to this point we have been using the exponent laws to simplify and evaluate expressions that have numeric bases, like $7^3$73 or $\left(2\times5\right)^4$(2×5)4. Now we are ready to combine our knowledge of exponent laws with our knowledge of algebra to manipulate expressions that contain algebraic terms with powers.

All this means is that we are replacing the specific number in the base with some letter of the alphabet. This letter is then called a variable.

For example, what do each of these numbers written in exponential notation have in common?

$2^5,3^5,11^5,736^5,9.02^5,\left(-0.18\right)^5,\pi^5,\left(6+\frac{2}{7}\right)^5$25,35,115,7365,9.025,(−0.18)5,π5,(6+27​)5

They all have a different base, but a common power of $5$5. This means we say that these numbers have the form $a^5$a5, where $a$a is an algebraic variable that we use to represent any number.

 

Worked example

Simplify $d\times d\times d\times d$d×d×d×d by writing the expression in exponential form.

Think: The variable $d$d occurs four times in the product, so we can use exponential notation to write it in the form of a base raised to a power.

Do: $d\times d\times d\times d=d^4$d×d×d×d=d4

Reflect: The simplified expression is $d^4$d4, where the variable $d$d is the base and the number $4$4 is the power. If we were given a particular value of $d$d, we could substitute it into the expression and evaluate it to get a particular number. For example, when $d=3$d=3, then $d^4=3^4=81$d4=34=81.

 

Multiple variables

We might see expressions that have more than one power, which we can still write in expanded notation. For example, $a^2b^4$a2b4 is the same as $a^2\times b^4$a2×b4, and so we can be expand the expression to get $a\times a\times b\times b\times b\times b$a×a×b×b×b×b.

 

Worked example

Write the following using exponential notation: $7\times u\times u\times2\times v\times v\times v\times v$7×u×u×2×v×v×v×v.

Think: Notice that the expression consists entirely of multiplication. To express in exponential form, we can find the number of times each base is multiplied by itself.

Do: There is one $7$7, which is written as $7^1$71 or just $7$7. The variable $u$u is multiplied two times. This can be written as $u^2$u2.

There is one $2$2, which is $2^1$21 or $2$2. The second variable $v$v is multiplied four times. This can be written as $v^4$v4.

So expressing in exponential form, we get:

$7\times u\times u\times2\times v\times v\times v\times v$7×u×u×2×v×v×v×v	$=$=	$7\times u^2\times2\times v^4$7×u2×2×v4
	$=$=	$14u^2v^4$14u2v4

Reflect: In the final line we evaluated the numeric product $7\times2$7×2, and removed the unnecessary multiplication symbols. Notice that the algebraic variables $u$u and $v$v are different, so we cannot combine them any further.

 

Worked example

Convert to expanded notation form: $11m^2\times5\left(-n\right)^3$11m2×5(−n)3.

Think: The first part of the product, $11m^2$11m2, is made up of the coefficient $11$11 and the base variable $m$m that is raised to the power of $2$2. Notice that this power is only acting on the $m$m, not on the $11$11. In the second part, $5\left(-n\right)^3$5(−n)3, the base is $\left(-n\right)$(−n).

Do:

$11m^2\times5\left(-n\right)^2$11m2×5(−n)2	$=$=	$11\times m\times m\times5\times\left(-n\right)\times\left(-n\right)\times\left(-n\right)$11×m×m×5×(−n)×(−n)×(−n)
	$=$=	$11\times m\times m\times5\times\left(-1\right)\times n\times\left(-1\right)\times n\times\left(-1\right)\times n$11×m×m×5×(−1)×n×(−1)×n×(−1)×n
	$=$=	$-1\times11\times m\times m\times5\times n\times n\times n$−1×11×m×m×5×n×n×n
	$=$=	$-55\times m\times m\times n\times n\times n$−55×m×m×n×n×n

Reflect: In the second line we took out $-1$−1 as a factor of each $\left(-n\right)$(−n), and since there were an odd number of them they multiply together to give $-1$−1. In the final line we evaluated the product of all the numeric terms. Notice that the three $n$ns in the expanded expression became positive, but the whole expression is negative.

 

Practice questions

Question 1

Question 2

Question 3