Which of the following is equivalent to 𝑥 to the negative two power times 𝑥 plus 𝑥 over 𝑥 plus 𝑥 plus 𝑥? A) Two over three 𝑥, B) one over 𝑥, C) three over two 𝑥, or D) two over three 𝑥 squared.
At this point, we have to remember what we do when we’re adding variables together. 𝑥 plus 𝑥 equals two 𝑥. Its exponent, its power, doesn’t change. If you can’t remember what to do here, try substituting numbers. If you had three plus three, that would equal two times three. It wouldn’t equal three squared. Once that squared away, we know that, in the denominator, 𝑥 plus 𝑥 plus 𝑥 equals three 𝑥. Bring down our 𝑥 to the negative two power. At this point, in our fraction, we have an 𝑥 over 𝑥. And those values cancel out, because 𝑥 over 𝑥 just equals one. So we have two-thirds times 𝑥 to the negative two power. 𝑥 to the negative two power can be written as one over 𝑥 squared. Multiply that by two-thirds. When we’re multiplying fractions, we multiply their numerators. One times two is two. And 𝑥 squared times three is three 𝑥 squared.
Two over three 𝑥 squared is equivalent to the expression we started with. And that’s option D.