Video Transcript
Which of the following expressions is equal to negative two to the negative eighth power times negative two to the negative sixth power? (A) Two to the 14th power, (B) two to the negative 14th power, (C) negative two to the negative 14th power, (D) four to the negative 14th power, or (E) negative two to the 48th power.
Looking at this expression, let’s see if we can simplify it. If we remember our exponent rules, we know that 𝑥 to the 𝑎-power times 𝑥 to the 𝑏-power equals 𝑥 to the 𝑎 plus 𝑏-power. In our expression, we are dealing with two values that have the same base. And that means we can simplify by adding their two exponent values together. But we should be careful with our negative signs here. We’re adding negative eight plus negative six so that we’d now have negative two to the negative 14th power. The key here is that we are taking negative two to the negative 14th power. This is not the same as negative two to the negative 14th power.
This is because negative two to the negative 14th power as it’s written in our answer choice is equal to negative one times two to the negative 14th power. And that is not what we have. We do have one option where negative two is in the parentheses, but it’s using the wrong exponents. So we need to try and rewrite what we have another time. One way to do that is to break up the negative two inside the parentheses into its factors. Negative two can be written as negative one times positive two. And when we have 𝑥 times 𝑦 to the 𝑎-power, we can distribute the 𝑎-power so that we have 𝑥 to the 𝑎-power times 𝑦 to the 𝑎-power.
In this case, we’d have negative one to the negative 14th power times two to the negative 14th power. And negative one to the negative 14th power is equal to one over negative one to the positive 14th power. And negative one to the positive 14th power, since 14 is even, will be equal to one so that we have one times two to the negative 14th power, two to the negative 14th power, which is option (B) here.
