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374 Mastering Technical Mathematics
Using the same numerical values as before, you can demonstrate this. Plug in the numbers
on the left-hand side of the equation and evaluate.
Now evaluate the right-hand side:
10 4 / 10 6 10,000 / 0.000001
10,000 · 1,000,000
10,000,000,000
10 10
10 [4(6)] 10 (46)
10 10
Here’s a more complicated property of exponentials. At least, it looks complicated until
you get used to working it out. Let x and y be real numbers, with the restriction that y
cannot be equal to 0. Then the exponential of x/y is equal to the exponential of 1/y with
base b x :
b (x/y) (b x ) (1/y)
Try an example where the base b is equal to 10, with exponents x 4 and y 7.
Evaluating the left-hand side first, letting 4/7 ≈ 0.5714, and using the “x y ” or “x^y”
function key on your calculator, you should get
10 (4/7) 10 0.5714
3.727
Alternatively, you can enter 0.5714, hit the “Inv” key, and then hit “log” to find 10 to
the power of 0.5714. Now plug the numbers into the right-hand side of the general
equation and work it out.
(10 4 ) (1/7) 10,000 (1/7)
To do this on your calculator, first figure 1/7 to four significant figures. You should get
0.1429. Then enter 10,000, hit the “x y ” or “x^y” key, and enter 0.1429. The result should
be 3.729. There’s a significant discrepancy here (2 parts in 10,000) because you’ve
taken a rounding error to the 7th power.
Exponentials can show the relationship between a “power of a power” and a product.
Suppose x and y are real numbers. Then the yth power of the exponential of x is equal
to the exponential of the product xy.
(b x ) y b (xy)