NZ Level 7 (NZC) Level 2 (NCEA)
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Division law with variable bases and negative powers
Lesson

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

$a^x\div a^y=a^{x-y}$ax÷​ay=axy

But what happens when the $y$y value is larger than the $x$x value? Well we subtract them in just the same way except we'll end up with a negative answer. For example, $a^2\div a^5=a^{2-5}$a2÷​a5=a25 which can be simplified to $a^{-3}$a3.

 

Examples

Question 1

Simplify the following, giving your answer in index form: $\frac{35x^2y^2}{-5x^{14}y^{14}}$35x2y25x14y14.

Think: Let's separate the terms and apply the Index division law.

Do

$35\div\left(-5\right)$35÷​(5) $=$= $-7$7 the coefficient term
$x^2\div x^{14}$x2÷​x14 $=$= $x^{2-14}$x214 the $x$x's
  $=$= $x^{-12}$x12  
$y^2\div y^{14}$y2÷​y14 $=$= $y^{2-14}$y214 the $y$y's
  $=$= $y^{-12}$y12  
$\frac{35x^2y^2}{-5x^{14}y^{14}}$35x2y25x14y14 $=$= $-7x^{-12}y^{-12}$7x12y12 all together

 

Question 2

Convert the following to a fraction and simplify using the index laws: $\left(-5x^6\right)\div\left(-3x^8\right)\div\left(-5x^6\right)$(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:

$\left(-5x^6\right)\div\left(-3x^8\right)\div\left(-5x^6\right)$(5x6)÷​(3x8)÷​(5x6) $=$= $\frac{-5x^6}{-3x^8}\times\frac{1}{-5x^6}$5x63x8×15x6
  $=$= $\frac{-5x^6}{15x^{14}}$5x615x14
  $=$= $\frac{-1}{3x^8}$13x8

 

Question 3

Convert the following to a fraction and simplify using the index laws:

$\left(-240u^{32}\right)\div\left(-8u^9\right)\div\left(-5u^{12}\right)$(240u32)÷​(8u9)÷​(5u12)

Question 4

Simplify the following, giving your answer with a positive index: $\frac{9x^3}{3x^{-4}}$9x33x4

 

Outcomes

M7-6

Manipulate rational, exponential, and logarithmic algebraic expressions

91261

Apply algebraic methods in solving problems

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