Video Transcript
Which of the following is equal to
two-thirds to the power of negative three? The options are (A) 27 over eight,
(B) eight over 27, (C) negative eight over 27, (D) negative six over negative nine,
or (E) negative 27 over eight.
So, to solve this problem, what
we’re gonna do is we’re going to use an adaptation of the most common exponent
rule. And that exponent rule is 𝑥 to the
power of negative 𝑛 is equal to one over 𝑥 to the power of 𝑛. Because, in fact, what this really
means is 𝑥 to the power of negative 𝑛 is equal to one to the power of 𝑛 over 𝑥
to the power of 𝑛. However, as one to the power of 𝑛
is just one, we write one over 𝑥 to the power of 𝑛. So therefore, if we’ve got 𝑥 over
𝑦 to the power of negative 𝑛, this is gonna be equal to 𝑦 to the power of 𝑛 over
𝑥 to the power of 𝑛. Cause what we do is we find the
reciprocal of our fraction and we put each of the terms raised to the power of
𝑛.
So therefore, if we got two-thirds,
or two over three, to the power of negative three, well, first of all, what we do is
we find the reciprocal. And the reciprocal is what we get
if we flip the numerator and denominator. So, two-thirds becomes three-halves
or three over two. Well, then, what we’re gonna do is
raise both the numerator and the denominator to the power of three because, in our
example, this is our 𝑛. Well, three cubed means three
multiplied by three multiplied by three, and two cubed means two multiplied by two
multiplied by two.
So therefore, we can say that
two-thirds to the power of negative three is gonna be equal to 27 over eight. And that’s because three times
three times three is 27 and two times two times two is eight. So therefore, we can say that the
correct answer from our list is going to be answer (A) because this is 27 over eight
or twenty-seven eighths.
