Multiplication law with integer bases
When multiplying a number by itself repeatedly, we are able to use index notation to write the expression more simply. Here we are going to look at a rule that allows us simplify products that involve the multiplication of index terms.
Consider the expression $a^5\times a^3$a5×a3. Notice that the terms share like bases.
Let's think about what this would look like if we expanded the expression:
We can see that there are eight $a$as being multiplied together, and notice that $8$8 is the sum of the powers in the original expression.
So, in our example above,
| $a^5\times a^3$a5×a3 | $=$= | $a^{5+3}$a5+3 |
| $=$= | $a^8$a8 |
Le's look at a specific example. Say we wanted to find the value of $4^2\times4^3$42×43. By evaluating each product separately we would have
| $4^2\times4^3$42×43 | $=$= | $16\times64$16×64 |
| $=$= | $1024$1024 |
Alternatively, by first expanding the terms in the original expression we can arrive at a simplified version of the expression on our way to the final value.
| $4^2\times4^3$42×43 | $=$= | $\left(4\times4\right)\times\left(4\times4\times4\right)$(4×4)×(4×4×4) |
| $=$= | $4^5$45 | |
| $=$= | $1024$1024 |
Notice in the second line we have identified that $4^2\times4^3=4^5$42×43=45.
The index law of multiplication
We can avoid having to write each expression in expanded form by using the multiplication law.
For any base number $a$a, and any numbers $m$m and $n$n as powers,
$a^m\times a^n=a^{m+n}$am×an=am+n
That is, when multiplying terms with a common base:
- Keep the same base
- Find the sum of the powers
When multiplying terms with like bases, we add the indices (or powers).
Multiplying like bases
The multiplication law only works for terms with the same bases.
Consider the expression $7^2\times3^2$72×32.
$7$7 and $3$3 are not the same base terms, so we cannot simplify this expression any further.
But we can simplify the following expression: $7^2\times7^4\times3^2$72×74×32.
Notice that two of the terms have like bases, so we can add their powers.
| $7^2\times7^4\times3^2$72×74×32 | $=$= | $7^{2+4}\times3^2$72+4×32 |
| $=$= | $7^6\times3^2$76×32 |
The power of $1$1
An index (or power) is telling us to multiply the base number by itself a certain number of times. For example:
| $5^4$54 | $=$= | $5\times5\times5\times5$5×5×5×5 |
| $5^3$53 | $=$= | $5\times5\times5$5×5×5 |
| $5^2$52 | $=$= | $5\times5$5×5 |
From this pattern we can see that $5^1$51 is the same as $5$5.
We can use this fact to simplify an expression like $9^2\times9^3\times9$92×93×9 by first writing it as $9^2\times9^3\times9^1$92×93×91 before applying the multiplication law.
| $9^2\times9^3\times9$92×93×9 | $=$= | $9^2\times9^3\times9^1$92×93×91 |
| $=$= | $9^{2+3+1}$92+3+1 | |
| $=$= | $9^6$96 |
Practice questions
Question 1
Simplify the following, giving your answer with a positive index: $2^2\times2^2$22×22
Question 2
Simplify the following, giving your answer in index form: $4\times5^6\times5^7$4×56×57.
Question 3
Simplify the following, giving your answer in index form: $9\times\left(-10\right)^4\times10^8$9×(−10)4×108.