Video: Multiplying Positive Mixed Number Expressions with Mixed Positive and Negative Integer Exponents

Which of the following is equal to (1(3/5))² × (1(3/5))⁻³? [A] 5/8 [B] 25/64 [C] 8/3 [D] −5/8 [E] −8/5

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Video Transcript

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.

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