Which of the following is equal to negative ten-ninths times 𝑥 to the negative two power times 𝑦 to the negative seven power? (A) Negative 10𝑥 squared 𝑦 to the seventh over nine, (B) negative 10 over nine 𝑥 to the seventh power 𝑦 squared, (C) negative nine over 10 times 𝑥 squared times 𝑦 to the seventh power, or (D) negative 10 over nine 𝑥 squared 𝑦 to the seventh power.
First of all, let’s think about what’s happening with the expression we’re given. We are multiplying negative ten-ninths times 𝑥 to the negative two power times 𝑦 to the negative seven power. One way to simplify this expression would be to rewrite 𝑥 to the negative two power and 𝑦 to the negative seven power as values where the exponents are positive. To do that, we remember our exponent rules that say 𝑥 to the negative 𝑎 power is equal to one over 𝑥 to the 𝑎 power. So, we can write 𝑥 to the negative two power as one over 𝑥 squared. In the same way, we can rewrite 𝑦 to the negative seven power as one over 𝑦 to the positive seventh power.
After that, we remember our rules for multiplying fractions together. We multiply their numerators and we multiply their denominators. In the numerator, we have negative 10 multiplied by one multiplied by one. And in the denominator, we have nine multiplied by 𝑥 squared multiplied by 𝑦 to the seventh power. And so, we could rewrite this expression as negative 10 over nine 𝑥 squared 𝑦 to the seventh power, which in this case is option (D).
Option (A) is not correct because they’ve rewritten the exponents as positive without moving them into the denominator. Option (B) has switched the exponents between the variable 𝑥 and 𝑦. And option (C) has flipped the fraction negative 10 over nine, which is not an equivalent fraction. Only option (D) produces an equal expression to what we started with.