Which of the following is equal to
one and three-fifths squared times one and three-fifths to the negative three
power? (A) Five-eighths, (B) 25 over 64,
(C) eight-thirds, (D) negative five-eighths, or (E) negative eight-fifths.
We copy down our expression. Before we get started, we need to
make one very important clarification. We have the rule that 𝑥 times 𝑦
squared is equal to 𝑥 squared 𝑦 squared. You might be tempted to write one
squared times three-fifths squared. This is not true. And that’s because the mixed number
one and three-fifths represents one plus three-fifths. And it does not represent one times
three-fifths. And since we can’t do that kind of
distribution, we need a new strategy to simplify.
We’ll need to convert these mixed
numbers into improper fractions. The improper fraction will be one
times five plus three, which is eight, over the original denominator of five. Both of these values are the same
mixed number, so they’re both the eight-fifths as an improper fraction. Once we’re to this point, since
these exponents have the same base, we can add their powers. Two plus negative three is negative
one. And from there, we are able to
distribute that power so that we have eight to the negative one power over five to
the negative one power.
Since we have a negative power in
the numerator and a negative power in the denominator, we can flip them. Five to the first power is
five. Eight to the first power is
eight. The equivalent expression is
five-eighths, which is option (A). The key to solving this one was
recognizing that you needed to change the format of these mixed numbers before you
went about solving.