We've already learnt the division law which states: $a^x\div a^y=a^{x-y}$ax÷ay=ax−y.
But what happens when $y$y is bigger than $x$x? For example, if I simplified $a^3\div a^5$a3÷a5 using the division law, I would get $a^{-2}$a−2. So what does a negative exponent mean? Let's expand the example to find out:
Remember that when we are simplifying fractions, we are looking to cancel out common factors in the numerator and denominator. Remember that any number divided by itself is $1$1.
So using the second approach, we can also express $a^3\div a^5$a3÷a5 with a positive exponent as $\frac{1}{a^2}$1a2.
We saw that $a^{-2}$a−2 is equivalent to $\frac{1}{a^2}$1a2.
More generally:
$a^{-x}=\frac{1}{a^x}$a−x=1ax
where $x$x is not equal to $0$0.
In other words, $a^{-x}$a−x is the reciprocal (i.e. the flipped version) of $a^x$ax. So $a^{-x}$a−x actually represents a fraction, NOT a negative number.
Consider the number represented by $2^{-3}$2−3. What is this number?
Well we can rewrite it in positive exponential form by taking the reciprocal: $2^{-3}=\frac{1}{2^3}$2−3=123
But $\frac{1}{2^3}=\frac{1}{8}$123=18. So we have found another way to represent the fraction $\frac{1}{8}$18: $2^{-3}=\frac{1}{8}$2−3=18
There are so many other fractions that can be represented in negative exponential form. What others can you think of?
What do we make of an expression such as $\left(\frac{5}{2}\right)^{-2}$(52)−2 or $\frac{4^{-3}}{5^{-1}}$4−35−1, where the numerator and denominator are both raised to negative powers?
Let's take it slowly through this $\left(\frac{5}{2}\right)^{-2}$(52)−2and see if we can come up with a shortcut.
First of all, $\left(\frac{5}{2}\right)^{-2}$(52)−2 can be split up in a couple of different ways. I am going to rewrite it as $\frac{5^{-2}}{2^{-2}}=5^{-2}\div2^{-2}$5−22−2=5−2÷2−2.
Now let's look at each part separately: $5^{-2}=\frac{1}{5^2}$5−2=152 $2^{-2}=\frac{1}{2^2}$2−2=122
Of course these fractions can be simplified but let's keep them in exponential form for now:
Let's see how far we've come from the original expression:
$\left(\frac{5}{2}\right)^{-2}$(52)−2 | $=$= | $5^{-2}\div2^{-2}$5−2÷2−2 | |
$=$= | $\frac{1}{5^2}\div\frac{1}{2^2}$152÷122 | (since dividing by a fraction is multiplying by its reciprocal) | |
$=$= | $\frac{1}{5^2}\times\frac{2^2}{1}$152×221 | ||
$=$= | $\frac{2^2}{5^2}$2252 | ||
$=$= | $\left(\frac{2}{5}\right)^2$(25)2 |
Compare $\left(\frac{2}{5}\right)^2$(25)2 and $\left(\frac{5}{2}\right)^{-2}$(52)−2. It seems that we can make the powers positive by switching the numerator and denominator around.
Simplify $\left(\frac{c}{d}\right)^{-4}$(cd)−4
Solution:
$\left(\frac{c}{d}\right)^{-4}=\left(\frac{d}{c}\right)^4$(cd)−4=(dc)4
Evaluate $2^{-3}$2−3 by first expressing it with a positive exponent. Give your answer as a fraction.
Simplify the following using exponent laws, giving your answer as a fully simplified fraction: $\left(\frac{5}{3}\right)^{-2}$(53)−2
Evaluate $5\times4^{-2}+9^0$5×4−2+90 without the use of a calculator.