Division law with variable bases and integer negative powers

We've already learnt about the division law which states:

$$ax÷​ay=ax−y

But what happens when the $$y value is larger than the $$x value? Well we subtract them in just the same way except we'll end up with a negative answer. For example, $$a2÷​a5=a2−5 which can be simplified to $$a−3.

 

Examples

Question 1

Simplify the following, giving your answer in exponential form: $$35x2y2−5x14y14​.

Think: Let's separate the terms and apply the Exponent Division law.

Do: 

$$35÷​(−5)	$$=	$$−7	the coefficient term
$$x2÷​x14	$$=	$$x2−14	the $$x's
	$$=	$$x−12
$$y2÷​y14	$$=	$$y2−14	the $$y's
	$$=	$$y−12
$$35x2y2−5x14y14​	$$=	$$−7x−12y−12	all together

 

Question 2

Convert the following to a fraction and simplify using the exponent laws: $$(−5x6)÷​(−3x8)÷​(−5x6).

Think: Dividing by a fraction is the same as multiplying by its reciprocal. We will then solve this problem in a couple of steps

Do:

$$(−5x6)÷​(−3x8)÷​(−5x6)	$$=	$$−5x6−3x8​×1−5x6​
	$$=	$$−5x615x14​
	$$=	$$−13x8​

 

Question 3

Question 4