2.03 Division properties of exponents and the zero exponent
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.
If we wanted to simplify the expression $a^6\div a^2$a6÷a2, we could write it as:
We can see that there are six $a$as being divided by two $a$as to give a result of four $a$as, and notice that $4$4 is the difference of the powers in the original expression.
So, in our example above,
| $a^6\div a^2$a6÷a2 | $=$= | $a^{6-2}$a6−2 |
| $=$= | $a^4$a4 |
Let's look at another specific example. Say we wanted to find the value of $2^7\div2^3$27÷23. By evaluating each term in the quotient separately we would have
| $2^7\div2^3$27÷23 | $=$= | $128\div8$128÷8 |
| $=$= | $16$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.
| $2^7\div2^3$27÷23 | $=$= | $\left(2\times2\times2\times2\times2\times2\times2\right)\div\left(2\times2\times2\right)$(2×2×2×2×2×2×2)÷(2×2×2) |
| $=$= | $2^4$24 | |
| $=$= | $16$16 |
Notice in the second line we have identified that $2^7\div2^3=2^4$27÷23=24.
The division property for exponents
We can avoid having to write each expression in expanded form by using the division property (which is also known as the quotient property).
$\frac{a^m}{a^n}=a^{m-n}$aman=am−n, where $a$a is any number,
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 $a^m\div a^n=a^{m-n}$am÷an=am−n.
Dividing terms with common bases
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 $\frac{9^8}{9^3}$9893 because the numerator and denominator have the same base: $9$9.
We cannot simplify $\frac{8^5}{7^3}$8573 because the two terms do not have the same base (one has a base of $8$8 and the other has a base of $7$7).
Worked example
Question 1
Simplify the following by first writing it in expanded form: $\frac{3^7}{3^2}$3732.
Think: We want to first write the expression in expanded form, so that we can then cancel out common factors from the numerator and denominator.
Do:
| $\frac{3^7}{3^2}$3732 | $=$= | $\frac{3\times3\times3\times3\times3\times3\times3}{3\times3}$3×3×3×3×3×3×33×3 |
| $=$= | $\frac{3\times3\times3\times3\times3}{1}$3×3×3×3×31 | |
| $=$= | $3^5$35 |
Reflect: We can see that the answer matches what we would expect if we used the division law. That is, $\frac{3^7}{3^2}=3^{7-2}$3732=37−2 giving us the final answer of $3^5$35.
Dividing coefficients
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.
Worked example
Question 2
Solve: Simplify $8x^6\div2x^4$8x6÷2x4 using exponent laws.
Think: First, let's write the expression as a fraction.
We then want to divide the coefficients (the numbers that are multiplied by the algebraic terms) and also use the division property, as we have a common base, and subtract the powers. Let's split the fraction up using the fact $\frac{a\times b}{c\times d}=\frac{a}{c}\times\frac{b}{d}$a×bc×d=ac×bd, to make the simplifications easier.
Do:
| $8x^6\div2x^4$8x6÷2x4 | $=$= | $\frac{8x^6}{2x^4}$8x62x4 |
| $=$= | $\frac{8}{2}\times\frac{x^6}{x^4}$82×x6x4 | |
| $=$= | $4\times\frac{x^6}{x^4}$4×x6x4 | |
| $=$= | $4\times x^2$4×x2 | |
| $=$= | $4x^2$4x2 |
Reflect: Combining the steps, we get $8x^6\div2x^4=4x^2$8x6÷2x4=4x2 and as this process becomes more familiar we can reduce the number of steps we take to arrive at the solution.
Zero power property
As with all of the other exponent laws, once we are familiar with working on numbers, we will apply the same properties of the zero power to bases that are algebraic variables or even algebraic expressions.
For any numeric or algebraic expression $a$a, the zero power property tells us that
$a^0=1$a0=1
Worked example
Question 3
Simplify $g^5\div g^5$g5÷g5 by first writing the expression in expanded form.
Think: This expression involves the division of one algebraic term by another. If we write it as a fraction we can see that the numerator and the denominator are the same.
Do:
| $g^5\div g^5$g5÷g5 | $=$= | $\frac{g^5}{g^5}$g5g5 | |
| $=$= | $\frac{g\times g\times g\times g\times g}{g\times g\times g\times g\times g}$g×g×g×g×gg×g×g×g×g | (We can cancel each similar factor in the numerator and denominator) | |
| $=$= | $\frac{1}{1}$11 | ||
| $=$= | $1$1 |
Reflect: Recall that $a^m\div a^n=\frac{a^m}{a^n}=a^{m-n}$am÷an=aman=am−n, so that we could have written $g^5\div g^5$g5÷g5 as $g^{5-5}=g^0$g5−5=g0. Now we can see that $g^0=1$g0=1. This should be no surprise; the initial expression asks us "What do we get when we divide $g^5$g5 by itself?", to which the answer is simply $1$1, since anything divided by itself is equivalent to $1$1.
Question 4
Simplify $\left(14p\right)^0$(14p)0.
Think: The presence of parentheses in the expression tells us that $14p$14p is the base and $0$0 is the power. The power acts on both the number $14$14 and the variable $p$p.
Do: The expression simplifies to $\left(14p\right)^0=1$(14p)0=1.
Reflect: Take note of where parentheses occur in an expression, as they can help us determine the correct order of operations when simplifying or evaluating. For example, in the expression $14p^0$14p0 the base is $p$p and the power is $0$0. Since $p^0=1$p0=1, this would simplify to $14\times1=14$14×1=14, which is different to the value of $\left(14p\right)^0$(14p)0 that we found above.
Combining exponent properties
We have come across a number of exponent laws up to this point: the multiplication property, the division property, the power of a power property and the zero power property.
Now we are going to look at questions that can be solved by using a combination of these rules. It's important to remember the order of operations when solving such questions.
We may also come across expressions of the form $\left(a^m\times b^n\right)^p$(am×bn)p, and we can use a combination of the multiplication and power of a power properties to see that
$\left(a^m\times b^n\right)^p=a^{m\times p}\times b^{n\times p}$(am×bn)p=am×p×bn×p.
Worked example
Question 5
Simplify $3p^5\times8p^2\div\left(6p^4\right)$3p5×8p2÷(6p4).
Think: Each important term in the expression has a coefficient, a base of $p$p, and some power. We also have a multiplication operation and a division operation. Considering the order of operations, we can simplify this expression by moving from left to right, using the multiplication law then the division law. Notice that $a\div\left(bc\right)$a÷(bc) is to be interpreted as $a\div\left(b\times c\right)$a÷(b×c), and not as $a\div b\times c=a\times c\div b$a÷b×c=a×c÷b.
Do:
| $3p^5\times8p^2\div\left(6p^4\right)$3p5×8p2÷(6p4) | $=$= | $3\times8p^5\times p^2\div\left(6p^4\right)$3×8p5×p2÷(6p4) | |
| $=$= | $24p^5\times p^2\div\left(6p^4\right)$24p5×p2÷(6p4) | (Perform numeric multiplication first) | |
| $=$= | $24p^{5+2}\div\left(6p^4\right)$24p5+2÷(6p4) | (By using the multiplication law) | |
| $=$= | $24p^7\div\left(6p^4\right)$24p7÷(6p4) | ||
| $=$= | $\frac{24p^7}{6p^4}$24p76p4 | (Rewrite as a fraction) | |
| $=$= | $\frac{4p^7}{p^4}$4p7p4 | (Perform numeric division first) | |
| $=$= | $4p^{7-4}$4p7−4 | (By using division law) | |
| $=$= | $4p^3$4p3 |
Reflect: In this example we were able to rearrange some of the terms in the expression because multiplication is commutative (the order doesn't matter). Depending on the values of the coefficients and the powers, some rearrangements might be more convenient for simplification than others. We can be creative with the exponent properties to see how many different ways we can simplify the same expression.
Another approach we could have used to solve the above expression is to first group the numeric factors and the variable factors separately, like so:
$3p^5\times8p^2\div6p^4=\left(3\times8\div6\right)\times\left(p^5\times p^2\div p^4\right)$3p5×8p2÷6p4=(3×8÷6)×(p5×p2÷p4)
Now the numeric part of the expression simplifies to $3\times8\div6=4$3×8÷6=4, and the variable part can be simplified by combining the multiplication law and the division law:
$p^5\times p^2\div p^4=p^{5+2-4}$p5×p2÷p4=p5+2−4
And so the final simplified form is $4p^{5+2-4}=4p^3$4p5+2−4=4p3, as expected.
Question 6
Evaluate the expression $\left(13m^4\right)^0+25\left(n^0\right)^8$(13m4)0+25(n0)8.
Think: This expression features powers of powers and zero powers. For each term in the sum, what is the base and what is the power?
Do:
| $\left(13m^4\right)^0+25\left(n^0\right)^8$(13m4)0+25(n0)8 | $=$= | $1+25\times1^8$1+25×18 | (By using the zero power law) |
| $=$= | $1+25\times1$1+25×1 | ($1$1 raised to any power is $1$1) | |
| $=$= | $1+25$1+25 | ||
| $=$= | $26$26 |
Reflect: The term $\left(13m^4\right)^0$(13m4)0 has a base of $13m^4$13m4 and a power of $0$0. Regardless how complex the base may be, if the power is $0$0 then we know the term will evaluate to $1$1.
Practice questions
Question 7
Fill in the box to make the statement true:
$15j^{14}\div\left(\editable{}\right)=5j^7$15j14÷()=5j7
Question 8
Simplify the following, giving your answer in exponential form: $\frac{3m^9n^4}{8m^8n^2}$3m9n48m8n2.
Question 9
Simplify the following, giving your answer with a positive exponent: $m^9\div m^5\times m^4$m9÷m5×m4