When multiplying a number by itself repeatedly, we are able to use exponent notation to write the expression more simply. Here we are going to look at a rule that allows us simplify products that involve the product of powers.

Exploration

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 distributed 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,

$a^5\times a^3$`a`5×`a`3	$=$=	$a^{5+3}$`a`5+3
	$=$=	$a^8$`a`8

We can avoid having to write each expression in expanded form by using the product of powers property.

Product of powers property

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 exponents

In other words, when multiplying terms with like bases, we add the powers.

Practice question

Question 1

Simplify the following, giving your answer in exponential form: $2^2\cdot2^3$22·23.

Question 2

Simplify the expression $8y^9\cdot5y^7$8y9·5y7.

Power of a power property

Now let's consider how we might rewrite expressions that have an exponential raised to another power.

Exploration

Consider the expression $\left(a^2\right)^3$(a2)3. What is the resulting power of base $a$a? To find out, have a look at the expanded form of the expression:

$\left(a^2\right)^3$(`a`2)3	$=$=	$\left(a^2\right)\times\left(a^2\right)\times\left(a^2\right)$(`a`2)×(`a`2)×(`a`2)
	$=$=	$\left(a\times a\right)\times\left(a\times a\right)\times\left(a\times a\right)$(`a`×`a`)×(`a`×`a`)×(`a`×`a`)
	$=$=	$a\times a\times a\times a\times a\times a$`a`×`a`×`a`×`a`×`a`×`a`
	$=$=	$a^6$`a`6

In the expanded form, we can see that we are multiplying six groups of $a$a together. That is, $\left(a^2\right)^3=a^6$(a2)3=a6.

We can confirm this result using the product of powers rule:

We know $\left(a^2\right)\times\left(a^2\right)\times\left(a^2\right)=a^{2+2+2}$(a2)×(a2)×(a2)=a2+2+2 which is equal to $a^6$a6.

We can avoid having to write each expression in expanded form by using the power of a power law.

The power of a power law

For any base number $a$a, and any numbers $m$m and $n$n as powers,

$\left(a^m\right)^n=a^{m\times n}$(am)n=am×n

That is, when simplifying a term with a power that itself has a power:

Keep the same base
Multiply the exponents

Practice question

Question 3

We want to simplify:

$\left(r^2\right)^4$(r2)4

Select the three expressions which are equivalent to $\left(r^2\right)^4$(r2)4:
$r^2\cdot r^4$r2·r4
A
$\left(r\cdot r\right)\cdot\left(r\cdot r\cdot r\cdot r\right)$(r·r)·(r·r·r·r)
B
$\left(r\cdot r\right)^4$(r·r)4
C
$\left(r\cdot r\right)\cdot\left(r\cdot r\right)\cdot\left(r\cdot r\right)\cdot\left(r\cdot r\right)$(r·r)·(r·r)·(r·r)·(r·r)
D
$r^2\cdot r^2\cdot r^2\cdot r^2$r2·r2·r2·r2
E
Choose the correct statement:
$\left(r^2\right)^4=r^{2+4}$(r2)4=r2+4
A
$\left(r^2\right)^4=r^{2\cdot4}$(r2)4=r2·4
B
Fill in the box to complete the rule: $\left(r^2\right)^4=r^{\editable{}}$(r2)4=r

Question 4

Express the following in simplified exponential form:

$\left(f^8\right)^6$(f8)6

Power of a product property

What if we want to convey that $8m$8m is the base, and the whole thing is being raised to some power? We can use parentheses to make this more clear, as outlined below.

Exploration

Let's think about the expression $5\times5\times5\times5\times a\times a\times a\times a$5×5×5×5×a×a×a×a. How can we write this in a more compact form, using our knowledge of exponent laws? Firstly, we can see that there are four lots of the number $5$5 and four lots of the variable $a$a being multiplied together. We can simplify the two parts separately like so

$5\times5\times5\times5\times a\times a\times a\times a=5^4\times a^4$5×5×5×5×a×a×a×a=54×a4.

That is a good start, but there is another way we can approach this. Using the fact that multiplication is commutative (the order that we multiply the numbers doesn't change the result) we can rearrange the product to get $5\times a\times5\times a\times5\times a\times5\times a$5×a×5×a×5×a×5×a. Notice that this expression has the same number of $5$5s and $a$as, just in a different order. Now if we look at groups of $5\times a$5×a, we can treat each one as a separate base and simplify the expression in the following way.

$5\times a\times5\times a\times5\times a\times5\times a$5×`a`×5×`a`×5×`a`×5×`a`	$=$=	$\left(5\times a\right)\times\left(5\times a\right)\times\left(5\times a\right)\times\left(5\times a\right)$(5×`a`)×(5×`a`)×(5×`a`)×(5×`a`)
	$=$=	$\left(5\times a\right)^4$(5×`a`)4

Using two different approaches, we have seen that $5\times5\times5\times5\times a\times a\times a\times a$5×5×5×5×a×a×a×a can be written as either $5^4\times a^4$54×a4 or $\left(5\times a\right)^4$(5×a)4, and this must mean that $5^4a^4=\left(5a\right)^4$54a4=(5a)4. In the second case, the base is "$5a$5a" and the power is $4$4.

As you might expect, we can use this specific example to arrive at a general rule about bases that are products of two numbers, or two variables, or a number and a variable.

Power of a product property

For the product of any numbers $a$a and $b$b in the base, and for any number $n$n in the power,

$\left(ab\right)^n=a^nb^n$(ab)n=anbn

In other words, a product raised to a power is equivalent to the product of the two factors each raised to the same power.

Careful!

Beware of the signs when raising negative numbers to a power, as a negative number raised to an odd power has a negative result. For example:

$\left(-2x^2\right)^3=-8x^6$(−2x2)3=−8x6

Practice question

Question 5

Simplify $\left(w^9v^6\right)^4$(w9v6)4.

Question 6

Simplify, and evaluate where possible, the following expression:

$\left(-3p^2\right)^5$(−3p2)5

Combining laws

In what situations could we use the multiplication law and the power of a power law together? Think about an expression like $\left(u^3\right)^5\times u^4$(u3)5×u4. The first term in the product, $\left(u^3\right)^5$(u3)5, has a base of $\left(u^3\right)$(u3) and a power of $5$5, and the second term, $u^4$u4, has a base of $u$u and a power of $4$4. At this point the two bases are different, so we cannot apply the multiplication law just yet.

One first step could be to rewrite the first term using the power of a power rule:

$\left(u^3\right)^5\times u^4$(`u`3)5×`u`4	$=$=	$u^{3\times5}\times u^4$`u`3×5×`u`4
	$=$=	$u^{15}\times u^4$`u`15×`u`4

Now both terms have a base of $u$u, so we can combine them by adding the powers: $u^{15+4}=u^{19}$u15+4=u19.

Practice questions

Question 7

Simplify the expression $8y^3\cdot9y^4$8y3·9y4.

Question 8

Simplify the following, giving your answer in exponential form: $5u^4v^2\cdot8u^3v^4$5u4v2·8u3v4.

Question 9

Simplify the following, giving your answer in exponential form: $\left(-8q^4\right)\cdot p^2\cdot\left(-7q^4\right)\cdot p^3$(−8q4)·p2·(−7q4)·p3.

Outcomes

MA.8.AR.1.1

Apply the Laws of Exponents to generate equivalent algebraic expressions, limited to integer exponents and monomial bases.

2.02 Multiplication properties of exponents

Product of powers property

Exploration

Practice question

Question 1

Question 2

Power of a power property

Exploration

Practice question

Question 3

Question 4

Power of a product property

Exploration

Practice question

Question 5

Question 6

Combining laws

Practice questions

Question 7

Question 8

Question 9

Outcomes

MA.8.AR.1.1

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