### Video Transcript

Which of the following has the same value as negative one and three-quarters to the power of negative three over negative one and three-quarters to the power of negative four? Is it (A) negative one and three-quarters to the power of negative seven? (B) Negative one and three-quarters to the power of seven. Is it (C) negative four-sevenths, (D) negative seven-quarters to the power of three-quarters, or (E) negative seven-quarters?

Here we have the quotient of two mixed numbers raised to a negative power. The first thing we’re going to do is change our mixed numbers into improper fractions. So we recall that to change one and three-quarters into an improper fraction, we begin by multiplying the integer by the denominator. That’s one times four, which is four. We then add the numerator, so we get four plus three, which is equal to seven. And so one and three-quarters is equal to seven-quarters.

This means that negative one and three-quarters is equal to negative seven-quarters. And so we rewrite the fraction in our question as negative seven-quarters to the power of negative three divided by negative seven-quarters to the power of negative four.

Next, we recall one of our index laws. And this says that when we divide two numbers whose base is equal, we subtract their exponents. So 𝑥 to the power of 𝑎 divided by 𝑥 to the power of 𝑏 is 𝑥 to the power of 𝑎 minus 𝑏. And so we get negative seven-quarters to the power of negative three minus negative four. Negative three minus negative four is negative three plus four, which is one. So we get negative seven-quarters to the power of one, which is simply negative seven-quarters.

And so the answer is (E). Negative one and three-quarters to the power of negative three over negative one and three-quarters to the power of negative four is simply negative seven-quarters.