Bruce-Alan M. answered • 11/14/12

Professor BAM (Tutors ALL of Long Island: Nassau + Suffolk)

An exponent following a value has 3 parts, which mean threee very different things:

1.) The denominator (on "top") is the "POWER" that the value is raised to (2 for squared, 3 for cubed, etc.)

2.) The numerator (or "bottom") is a "ROOT" (1/2 is square-root, etc.)

3.) A negative sign means "RECIPROCAL" (one over the value).

Thus, the exponent is -POWER/ROOT and a minus sign means RECIPROCAL.

Note that these three operations can be applied to the value in any order,

and if the value is itself a fraction these operations can be applied separately to the numerator and denominator (or simply the "top" and "bottom") of that value, so (TOP/BOTTOM)^POWER/ROOT is the same as TOP^(POWER/ROOT) / BOTTOM^(POWER/ROOT), then flip the fraction if there is a minus sign.

-----

In this case, the value to be raised to the -2/5 power is .00032, which is 32/(10,000) or 32 x 10^(-5),

so we are looking for (32/10000) ^ -2/5. Applying the -2/5 exponent separately to the top and bottom gives us (32^-2/5) / (10000^-2/5)

Since the order doesn't matter, it is usually easier to apply the power first (to get a bigger number before taking the root), so this would be: 32^2 / 10,000^2 or 1024 / 10^10. The next step would be to take the fifth root of each part ("top" and "bottom"), then flip the fraction upside-down to get the reciprocal. The top (before flipping) is the fifth root of 1024, which is 4 (because 4^5 is 1024); the bottom becomes 10^2, since (10^10)^1/5 is the same as 10^(10/5). So, we have 4/100, and the reciprocal is 100/4 or 25.

In this case, however, you might notice that 32 is really 2^5 and 10,000 is really 10^5, which might make it easier to take the fifth root first! Applying the second rule first makes the process much easier: instead of raising (32/10000) to the -2/5 power, we can raise (2^5/10^5) to that power. Applying the power separately gives us (2^5)^-2/5 on top and 10^5)^-2/5 on the bottom. Now, remember that to raise a power to a power, you MULTIPLY the exponents, so the top becomes 2^-(5*2/5) or just 2^-2, and the bottom becomes 10^-(5*2/5) or just 10^-2. So, we have 2^-2 / 10^-2 or (removing th minus powers by flipping) simply 10^2 / 2^2 or 100/4.

*** There is no "right" way to do these problems, since the operations are independent of each other ("orthagonal") and may be applied in any order, and each part of the fraction (top and bottom) can be treated separately. Furthermore, you should ALWAYS look for "coincidences" (like 2^5 and 10^5), especially on "standardized" tests (like the SAT), since the test makers deliberately put in these shortcuts to reward students who find them! This kind of problem is tedious, because there are several separate things to keep track of, but each step is very simple and it becomes mostly a matter of "keeping track" of details and perfoming very simple operations at the right time.