# Multiplication law with integer bases

Lesson

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.

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.

### Outcomes

#### NA5-2

Use prime numbers, common factors and multiples, and powers (including square roots)