Video Transcript
True or false: The simplified form of two 𝑥 to the power of negative two over 𝑦 to the power of negative three is two 𝑦 to the power of three over 𝑥 to the power of two.
The second expression can also be read as two 𝑦 cubed over 𝑥 squared. In order to answer this question, we need to recall one of our rules when dealing with negative exponents. We know that 𝑎 to the power of negative 𝑥 is equal to one over 𝑎 to the power of 𝑥. This means that the numerator of our expression, two 𝑥 to the power of negative two, is the same as two over 𝑥 squared or 𝑥 to the power of two. Our rule also means that one over 𝑎 to the power of negative 𝑥 is equal to 𝑎 to the power of 𝑥. As 𝑦 to the power of negative three is the denominator, this is the same as one over 𝑦 to the power of negative three. This must therefore be the same as 𝑦 to the power of three or 𝑦 cubed. The expression two 𝑥 to the power of negative two over 𝑦 to the power of negative three can therefore be rewritten as two over 𝑥 squared multiplied by 𝑦 cubed.
We know that 𝑦 cubed is the same as 𝑦 cubed over one. And multiplying the numerators gives us two 𝑦 cubed, and multiplying the denominators gives us 𝑥 squared. The expression simplifies to two 𝑦 cubed over 𝑥 squared. We can therefore conclude that the statement is true. Two 𝑥 to the power of negative two over 𝑦 to the power of negative three is the same as two 𝑦 cubed over 𝑥 squared.
