Division law with integer and variable bases

The division law

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).

 

Negative 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 $xx<y, the resulting power will be negative.

 

Dividing terms with common bases

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).

 

Dividing Coefficients

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

Examples

Question 1

Simplify the following using index 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

 

Question 2

Question 3