Video Transcript
Which of the following is equal to negative three and a half to the power of negative two? Is it (A) four over 49, (B) negative three and one-quarter, (C) 49 over four, (D) 14 over negative four, or (E) six and one-half?
In this question, we have a mixed number that’s being raised to a negative power. Before we can evaluate this, we need to change our mixed number into an improper fraction. We know that three and a half is equal to seven over two. We achieve this by multiplying the integer by the denominator — that’s three times two, which is six — and then adding the numerator, which is seven. The denominator stays the same. So three and a half is equal to seven over two. This means negative three and a half is equal to negative seven over two. And so negative three and a half to the power of negative two is negative seven over two to the power of negative two.
Next, we recall what we do when we have a negative exponent. We know that 𝑥 to the power of negative 𝑎 is one over 𝑥 to the power of 𝑎. A negative exponent means we need to take the reciprocal. The reciprocal of negative seven over two is negative two over seven. So negative seven over two to the power of negative two is negative two over seven squared. Then, to square our fraction, we simply square the numerator and separately square the denominator. We also know that a negative number squared is positive. So we get two squared, which is four, over seven squared, which is 49.
And we see that negative three and a half to the power of negative two is equal to four over 49. That’s (A).
