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Division law with variable bases


Previously we looked at how to divide power terms with numeric bases using the index law of division, or the division law.

Division Law

$\frac{a^m}{a^n}=a^{m-n}$aman=amn, 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.

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).




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).


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

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

Solve: Simplify $8x^6\div2x^4$8x6÷​2x4 using index 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 law, 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.


$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 amount of steps we take to arrive at the solution.

Practice questions

Question 1

Simplify the following, giving your answer in index form:


Question 2

Fill in the box to make the statement true:

  1. $15j^{14}\div\left(\editable{}\right)=5j^7$15j14÷​()=5j7

Question 3

Simplify the following, giving your answer in index form: $\frac{3j^5k^9}{4j^4k^6}$3j5k94j4k6.



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