Ratio in an exponent
Here’s a more complicated property of exponentials. Let x and y be real numbers, with the
restriction that y cannot be equal to 0. Then
b(x/y)= (bx)(1/y)whenb > 0. Let’s try an example where the base b is 10, with exponents x= 4 and y= 7.
Evaluating the left side first, letting 4/7 ≈ 0.5714 and using the “xy” or “x^y” function key on
a calculator, we get
10 (4/7)≈ 10 0.5714
≈ 3.727
Alternatively, we can enter 0.5714, hit the “Inv” key, and then hit “log” to find 10 to the
power of 0.5714. Now when we plug the numbers into the right side of the general equation
and work it out, we obtain
(10^4 )(1/7)= 10,000(1/7)
To work this out on a calculator, we must first figure 1/7 to four decimal places. That gives
us 0.1429. Then, we enter 10,000, hit the “xy” or “x^y” key, and enter 0.1429. The result is
3.729. There’s a discrepancy, because we’ve taken a rounding error to the seventh power!
Power of a power vs. product
Exponentials can show the relationship between a “power of a power” and a product. Suppose
thatx and y are real numbers. Then
(bx)y=b(xy)whenb > 0. To demonstrate this, let b=e,x= 2, and y= 3. Let’s evaluate the left side first, using
2.718 as the value of e and going to three decimal places during the calculation process:
(e^2 )^3 ≈ (2.718^2 )^3≈ 7.388^3
≈ 403.256
Now the right side:
e(2×3)=e^6≈ 2.718^6
≈ 403.178
How Exponentials Work 493