$\frac{a^x}{a^y}=a^{x-y}$axay=ax−y
It is similar to the multiplication law, however in this case we subtract the powers from one another, rather than add them. Let's look at an expanded example to see why this is the case.
If I wanted to simplify the expression $a^6\div a^2$a6÷a2, I could write it as:
Once I cancel out the common factors, I am left with $a^4$a4 (which is the difference between the two powers).
Consider a case where the power in the denominator is greater than the power in the numerator, for example $x^3\div x^5$x3÷x5
If we apply the division law, we get $x^{3-5}=x^{-2}$x3−5=x−2. We are left with a negative power.
Going back to the generalised form of the division law:
$\frac{a^x}{a^y}=a^{x-y}$axay=ax−y
If $x
As with using the multiplication (or product) law, you can only apply the division (or quotient) law to terms with the same bases (just like we can only add and subtract like terms in algebra).
We can simplify $\frac{x^5}{x^3}$x5x3 because the numerator and denominator have the same base $x$x.
We CANNOT simplify $\frac{x^5}{y^3}$x5y3 because the two terms DO NOT have the same base (one is base $x$x and the other is base $y$y).
We still follow a two-step process to simplify expressions with coefficients. Consider $8x^6\div2x^4$8x6÷2x4
First, we are going to divide the coefficients (i.e. the numbers that are multiplied by the algebraic terms):
Divide $8$8 by $2$2, to get $4$4
Second, using the division law, we subtract the powers.
$x^6\div x^4$x6÷x4 | $=$= | $x^{6-4}$x6−4 |
$=$= | $x^2$x2 |
Combining the two steps, we get $8x^6\div2x^4=4x^2$8x6÷2x4=4x2
Simplify the following using exponent laws: $\frac{3^7}{3^2}$3732
Solution:
Since the bases are the same we can apply the Division Law:
$\frac{3^7}{3^2}$3732 | $=$= | $3^{7-2}$37−2 |
$=$= | $3^5$35 |
Evaluate the following using exponent laws: $5\times\frac{4^5}{4^3}$5×4543.
Evaluate
$\frac{4}{7}\times3^4\div3^6$47×34÷36