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I am seriously questioning my sanity right now, I have been trying to find a solution to this. My only conclusion right now, is that this must be a mistake in this algebra book.

This should be a very basic thing to know right?

1 / (1/(1/3^4)) cannot be equal to 1 * (3^4/1) can it???

I even checked it in my calculator, the first one is equal to 1/81 and the other is equal to 81/1, so clearly, they are not equal right? [![enter image description here][1]][1]

https://i.stack.imgur.com/48phb.jpg

postmortes
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Mobius-User
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2 Answers2

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When you use slashes for division it is not clear where the precedence is. The official convention, discussed many places on this site, is that you compute from left to right, so $1/1/1/3^4=((1/1)/1)/3^4=(1/1)/3^4=1/3^4=3^{-4}$ The exponent binds more tightly than division. Your book, however, has used stacked fractions, which come with implied parentheses. $\dfrac 1{\frac 1{3^4}}=1/(1/3^4)=1/3^{-4}=3^4$. The grouping is the cause of the disagreement.

Ross Millikan
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  • Yes. Note that the first has $1$ divided by $\frac 1{3^4}$ and the second has $1$ times $\frac {3^4}1$. In the typed portion of your question you have an extra $1/$ on top beyond what the book has. That is the source of the error. Note that 1 / (1/(1/3^4)) has three ones while the book has only two. – Ross Millikan Jun 19 '17 at 19:06
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The math is correct. One thing to look out for with a calculator is that you have to put parentheses in the right place in order to get the right answer:

$$\frac{1}{3^{-4}} = 1/(1/(3^4))$$

If you just type in $1/1/3^4$, your calculator might compute $(1/1)/3^{4}$ which will give you a different (incorrect) answer.

One way to see why exponents work this way is to remember two rules from algebra:

$$\frac{1}{a}\times a = 1$$ $$b^{-k} \times b^{k} = 1.$$

Based on the first equation, we know that $\frac{1}{3^{-4}}$, whatever it is, must be some number you can multiply by $3^{-4}$ to get 1:

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

Based on the second equation, we know that

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

If you notice what is common between these two equations, you can see that $$\frac{1}{3^{-4}} = 3^{4}.$$

But actually, in my daily life, I usually just check my math according to these rules: (1) positive exponents make large numbers larger and small numbers smaller (2) in contrast, negative exponents make large numbers small, and make small numbers large. (3) reciprocals do the same thing that negative exponents do.

So $$\frac{1}{3^{-4}}$$ will change $3^{-4}$ from a big to a small number or vice-versa. $3^{-4}$ is a small number, because of the negative exponent. Hence $1/{3^{-4}}$ will be a big number, like $3^{4}$, rather than a small number like $3^{-4}$.

user326210
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  • @Novantix Sure: the general rule is $a \div \frac{b}{c} = a \times \frac{c}{b}.$ Another way to see this is to make $a$ into the fraction $(a/1)$. $a \div \frac{b}{c} = \frac{a}{1}\div \frac{b}{c} = \frac{a}{1}\times \frac{c}{b}$ --- to divide by a fraction, you multiply by the reciprocal. – user326210 Jun 19 '17 at 19:11