2.06 Division property of exponents
Introduction
The method to divide power terms is similar to the multiplication property , however in this case we subtract the powers from one another, rather than add them. Let's look at an example where we distribute the terms to see why this is the case.
The division property for exponents
If we wanted to simplify the expression a^{6} \div a^{2}, we could write it as:
We can see that there are six a's being divided by two a's to give a result of four a's, and notice that 4 is the difference of the powers in the original expression.
So, in our example above,
| \displaystyle a^{6} \div a^{2} | \displaystyle = | \displaystyle a^{6-2} |
| \displaystyle = | \displaystyle a^{4} |
Let's look at another specific example. Say we wanted to find the value of 2^{7} \div 2^{3}. By evaluating each term in the quotient separately we would have
| \displaystyle 2^{7} \div 2^{3} | \displaystyle = | \displaystyle 128 \div 8 |
| \displaystyle = | \displaystyle 16 |
Alternatively, by first distributing the terms in the original expression we can arrive at a simplified version of the expression on our way to the final value.
| \displaystyle 2^{7} \div 2^{3} | \displaystyle = | \displaystyle \dfrac{2\times 2\times 2\times 2\times 2\times 2\times 2}{2\times 2\times 2} |
| \displaystyle = | \displaystyle 2^{4} | |
| \displaystyle = | \displaystyle 16 |
Notice in the second line we have identified that 2^{7}\div 2^{3}=2^{4}
We can avoid having to write each expression in expanded form by using the division property (which is also known as the quotient property): \dfrac{a^{m}}{a^{n}}=a^{m-n}, where a is any base number, and m and n as powers.
That is, when dividing terms with a common base:
Keep the same base
Find the difference in the power.
Of course, we can also write this property in the form a^{m}\div a^{n} = a^{m-n}.
As with using the multiplication (or product) property, we can only apply the division (or quotient) property to terms with the same bases (just like we can only add and subtract like terms in algebra). We can simplify \dfrac{9^{8}}{9^{3}} because the numerator and denominator have the same base: 9.
We cannot simplify \dfrac{8^{5}}{7^{3}} because the two terms do not have the same base (one has a base of 8 and the other has a base of 7).
To simplify expressions with coefficients we follow the same steps as when we are multiplying expressions with coefficients. That is, we can treat the problem in two parts. Let's take a look at an example.
Examples
Example 1
Simplify the following: \dfrac{3^{9}\times2^{4}}{3^{8}\times2^{2}}
Example 2
Fill in the box to make the statement true: 15^{14} \div (⬚)=15^{7}
The division property for exponents:\dfrac{a^{m}}{a^{n}}=a^{m-n} where a is any base number, and m and n are powers.
That is, when dividing terms with a common base:
Keep the same base
Find the difference in the power.