How do you simplify (12x^6)/(6x^ -2)?

Jan 5, 2016

$\frac{12 {x}^{6}}{2 {x}^{- 2}} = 2 {x}^{8}$

Explanation given in detail. The method should help you deal with any question of this type.

Explanation:

Splitting this into 2 parts; numbers and variables

$\textcolor{b l u e}{\text{Part 1: The numbers") color(white)(".....}} \frac{12}{6}$

Divide top and bottom by 6 as $12 \div 6$ gives a whole number answer.

$\frac{12 \div 6}{6 \div 6} = \frac{2}{1} = 2$
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$\textcolor{b l u e}{\text{Part 2 : The variables") color(white)(".....}} \frac{{x}^{6}}{{x}^{-} 2}$

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$\textcolor{b r o w n}{\text{To explain what is happening: }}$

Suppose we had a different variable, say $z$.

If this is written as ${z}^{- 2}$ then it is another way of writing $\frac{1}{{z}^{2}}$
If this is written as $\frac{1}{{z}^{- 2}}$ then it is another way of writing ${z}^{2}$

So to summarise: If written this way in the numerator it is actually the denominator. On the other hand, if it is written this way in the denominator it is actually the numerator.

$\textcolor{g r e e n}{\text{You 'move it' to the other side of the dividing line.}}$
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Write ${x}^{6} / \left({x}^{- 2}\right) \text{ as } {x}^{6} \times {x}^{2}$

But ${x}^{6} \times {x}^{2}$ is the same value as ${x}^{\left(6 + 2\right)} = {x}^{8}$

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$\textcolor{b l u e}{\text{Putting it all together}}$

$\frac{12 {x}^{6}}{2 {x}^{- 2}} = 2 {x}^{8}$

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Another way of dealing with $\frac{1}{x} ^ \left(- 2\right)$

Multiply by 1 in the form of$\textcolor{w h i t e}{. .} {x}^{2} / {x}^{2}$ giving:

$\frac{1}{x} ^ \left(- 2\right) \times {x}^{2} / {x}^{2}$

$= \frac{1 \times {x}^{2}}{{x}^{- 2} \times {x}^{2}}$

$= {x}^{2} / \left({x}^{\left(- 2 + 2\right)}\right)$

${x}^{2} / {x}^{0}$

But ${x}^{0} = 1$ giving

${x}^{2} / 1 = {x}^{2}$