Which of the following is equal to
two to the negative two power times 27 to the negative one-third power times 25 to
the negative one-half power over four to the negative three power. A) One over 240, B) 15 over 16, C)
16 over 15, D) 48 over five, or E) 240
One thing that we notice almost
immediately is that all four of these exponent values are negative. And we know that 𝑥 to the negative
𝑎 power is equal to one over 𝑥 to the 𝑎 power. That is, when you have a negative
exponent in the numerator, you can move that exponent to the denominator and make it
positive. And by extension, we can say one
over 𝑥 to the negative 𝑎 power equals 𝑥 to the positive 𝑎 power.
In our case, it means all the
values with negative exponents in the numerator should be moved to the
denominator. And they will be positive. Remember, this is only possible
because we are multiplying all three of these values together. We now have in the denominator two
to the positive two power times 27 to the positive one-third power times 25 to the
positive one-half power. And four to the negative three
power can be moved to the numerator as four to the positive three power.
If we keep our numerator the same,
we can calculate two squared, which is four. And then, we have to remember that
27 to the one-third power is equal to the cube root of 27. It’s saying that some value cubed
equals 27. And if 𝑥 cubed equals 27, then 𝑥
equals 27 to the one-third power, or the cube root of 27. We know that three times three
times three equals 27; three cubed equals 27. And that means the cube root of 27
is three. And so, in place of 27 to the
one-third power, we can write three.
After that, we have the square root
of 25. If 𝑥 squared equals 25, then 𝑥
equals 25 to the one-half power, which is the same thing as the square root of
25. We know the square root of 25 is
five because five times itself is 25.
There is one more thing we can
simplify. If we look at our four in the
denominator, we could say that that is four to the first power. And since we have four cubed in the
numerator and four to the first power in the denominator, we can simplify that to
have four squared in the numerator. This is because when the exponents
have the same base, 𝑥 to the 𝑎 power over 𝑥 to the 𝑏 power equals 𝑥 to the 𝑎
minus 𝑏 power. In our case, we had four cubed over
four to the first power. And that is four to the three minus
one power, four squared. Three times five is 15. Four squared is 16. So, this simplification is 16 over
15, which is option C.