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I am a bit confused with negative exponents so here is an example.

Let $4$ to the $-3 = 1\div4\div4\div4 $

The thing which confuses me is the $1$, why we use $1$ before dividing. so if i do it this way

$4$ to the $-3 = 4\div4\div4$

why is it wrong ?

and my other question is why any negative number times negative number gives positive answer. so please give any simple explanation as if i am child.

rschwieb
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abdul raziq
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  • Sorry @bryansis2010 for rolling back: usually for a poster at this level we avoid altering the original notation too much. It's better to leave such things as-is to prevent confusion. That said, I will gladly roll myself back if abdul says the division signs were even better. Abdul, you can see the suggested edit by clickin gthe edit timestamp. – rschwieb Mar 06 '14 at 17:25
  • i think the edit was helpful. – abdul raziq Mar 06 '14 at 17:28
  • OK :) gladly re-reverting myself then! – rschwieb Mar 06 '14 at 17:29
  • cheers! i would say that when i was a beginner (mind that i got 101 rep due to Stackoverflow), questions without Latex was hard to understand and Latex was "cool"...so to this poster, i encourage to use Latex as this is the language of this qna! – bryan.blackbee Mar 06 '14 at 17:32

3 Answers3

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Since $4^0 = 1$, and $4^a 4^b = 4^{a+b}$, what happens if we set $b = -a$? We should get $$4^a 4^{-a} = 4^{a-a} = 4^0 = 1.$$

Now, suppose you have $4^3 \times 4^{-3}$. We know that this is $4^{3-3} = 4^0 = 1$, but what does it look like when multiplied through?

$$4^3 = 4\times 4\times 4.$$

To make this 1, we have to divide 1 by 4 three times:

$$4^{-3} = \frac{1}{4}\times \frac14 \times \frac14$$

which makes

$$4^3\times 4^{-3} = 4\times 4\times 4 \times \frac14\times \frac14\times \frac14 = 1.$$

Finally, note that $\frac14\times\frac14 = \frac{1}{4 \times 4} = \frac{\frac{1}{4}}{4}$, which is admittedly a poor way to write it.

In the end, think of negative exponents as "one over" the positive exponent: $$4^{-3} = \frac{1}{4^3}.$$

MPW
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Emily
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$4^{-3}$ is the same as $\frac{1}{4^3}=\frac{1}{4\cdot 4\cdot 4}=\frac14\cdot\frac14\cdot\frac14=(\frac14)^3=4^{-1}\cdot4^{-1}\cdot4^{-1}$. The

It is not clear to write ""1/4/4/4"". For one thing, this expression is ambiguous as to what order of operations is intended. It could be (1/4)/(4/4)=1/4, or else 1/((4/4)/4)=4. Actually $4^{-3}$ is a third order, which gives an even different answer: 1/64.

It seems like you are thinking $4^{-3}$ as "dividing by 4 three times," which is right, but you would have to be more careful writing it. To express it this way, you would say (((1/4)/4)/4), that is "divide the whole by four, then divide that by four again, then divide that by four again. Or "one quarter of one quarter of one quarter" if you like.

If you are comfortable with this statement: "dividing by four is the same as multiplying with $\frac14$", then you can see why $4^{-3}$ is the same as $\frac14\cdot\frac14\cdot\frac14$.

$4^{-1}$ is the same as $1/4$: it is one fourth of a whole. As yourself "why do we need the 1 in 1/4?"

rschwieb
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To answer your last question. Suppose your provision of beer bottles in the fridge goes down with a ratio of two bottles a day. To know how many bottles you'll have left tomorrow you take the number of bottles you have today, say $14$, and subtract $2$, you get $14-2=12$. To know how your provision will change in three days you multiply $3$ by $-2$, and get $-6$. You'll have $6$ bottles less than today. To know how your provision has changed in the last $4$ days you multiply $-4$ (that is: 4 days ago) by $-2$, and you get $8$: four days ago you had $8$ bottles more than today.

alex
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