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,
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
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.
Notice in the second line we have identified that $4^2\times4^3=4^5$42×43=45.
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,
That is, when multiplying terms with a common base:
When multiplying terms with like bases, we add the indices (or powers).
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.
An index (or power) is telling us to multiply the base number by itself a certain number of times. For example:
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.
Simplify the following, giving your answer with a positive index: $2^2\times2^2$22×22
Simplify the following, giving your answer in index form: $4\times5^6\times5^7$4×56×57.
Simplify the following, giving your answer in index form: $9\times\left(-10\right)^4\times10^8$9×(−10)4×108.