### Video Transcript

Which of the following expressions has the same value as negative 33 and a third to the power of three all squared? Is it A) negative 100 over three to the power of six? B) Negative 100 over three to the power of negative six. C) Negative 100 over three to the power of five. D) Negative 100 over three to the power of negative five. Or E) Negative 100 over three to the power of 32.

Despite the fact that all five options contain the improper or top-heavy fraction negative 100 over three, it is worth considering how we’ve changed the mixed number negative 33 and one-third to negative 100 over three. Let’s consider the fraction 33 and one-third.

In order to convert a mixed fraction into an improper fraction, we need to multiply the whole number by the denominator and then add the numerator. In this case, this is 33 multiplied by three plus one. 33 multiplied by three is equal to 99. And 99 plus one equals 100. This means that 33 and one-third is equal to one-hundred thirds or 100 over three. This means that we can rewrite negative 33 and one-third cubed all squared as negative 100 over three cubed all squared.

We now need to consider what happens to the exponents or indices. 𝑥 to the power of 𝑎 raised to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 multiplied by 𝑏. This law or rule tells us that we can multiply the exponents or indices. As three multiplied by two is equal to six, our expression can be rewritten as negative 100 over three to the power of six.

This means that option A, negative 100 over three to the power of six, has the same value as negative 33 and one-third to the power of three all squared.