Energy and Human Ambitions on a Finite Planet, 2021a

A.6 Fractional Powers 366
writing this set three times, all multiplied together, or (7 · 7 · 7 · 7) ×
(7 · 7 · 7 · 7) × (7 · 7 · 7 · 7), which is just 12 sevens multiplied, or 7 12 .So
we have discovered/formulated the rule:
(x a ) b x a·b . (A.7)
We multiply the exponents when raising the inner exponent to an outer
one.
How about 3 2 · 3 5 ? What is the rule there? The process 21 is similar to
before, exp<strong>and</strong>ing out to (3 · 3) × (3 · 3 · 3 · 3 · 3), which just looks like
seven threes multiplied together, or 3 7 . Therefore, our rule is:
x p · x q x (p+q) .
(A.8)
21: Note that some care must be exercised
in selecting the example. For instance, picking
both exponents as 2 would leave some
ambiguity: is the result of 4 2 + 2, 2 × 2,or
2 2 ?
We add exponents when multiplying two pieces, each having their own
exponent. Note that this does not work when the bases are unequal, as
you could verify yourself for 3 2 · 5 4 .
Finally, what about inversion, or dividing by x n ? As a preview, a negative
power is equivalent to putting the item in the denominator, so that
x −1 1 x
. To see this, consider Eq. A.8 in the case where p <strong>and</strong> q are
opposite sign but the same magnitude. For instance, following the “add
the exponents rule” we get that 3 4 ·3 −4 3 4−4 3 0 1, because anything
raised to the zero power is 1. 22 The only thing we can multiply into
1
3 · 3 · 3 · 3 in order to get 1 is
3·3·3·3 . This means that 3−4 is the same as
1/3 4 , or more generally:
1
x n x−n .
(A.9)
Negative exponents therefore flip the construction to the denominator,
or denote a division rather than multiplication.
22: Think of the exponent as how many instances
of a number are multiplied together
in a chain, implicitly all multiplied by 1. If
we have zero instances of the number, then
the implicit 1 is all we have left in the multiplication.
In other words, 1 is the starting
point for all multiplications, just like zero
is the starting point for all additions.
A.6 Fractional Powers
In the previous section, we only dealt with integer powers, so that we
could write out 3 4 as 3 · 3 · 3 · 3. How would we possibly write 3 1.7 ?Yet
it is mathematically well defined. A calculator has no trouble.
We can get a hint from Eq. A.8. Consider, for example, 5 1 2 · 5 1 2 . We know
that we can just add the exponents, which in this case add to a tidy 1,
meaning that the answer is just 5. Therefore we interpret 5 1 2 as the square
root of 5, since multiplying it by itself yields 5. So we can re-express our
familiar friend as a fractional power:
x 1 2
√
x. (A.10)
In principle, then, we could approach 3 1.7 by taking the tenth-root of 3
<strong>and</strong> raising it to 17 th power: 3 1.7 (3 1 10 ) 17 3 17
10 .
© 2021 T. W. Murphy, Jr.; Creative Commons Attribution-NonCommercial 4.0 International Lic.;
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