Calculate negative one-third squared multiplied by three-halves squared, giving your answer in its simplest form.
There are several ways of approaching this problem. One way would be to square each of the two terms and then multiply the two answers. Squaring any term means multiplying it by itself. Therefore, negative one-third squared is equal to negative one-third multiplied by negative one-third. When multiplying fractions, we multiply the two numerators and then separately the two denominators. Negative one multiplied by negative one is equal to one, as multiplying two negative numbers gives us a positive answer. Three multiplied by three is equal to nine. Therefore, negative one-third squared is one-ninth.
We can square three over two or three-halves using the same method. We multiply three over two by three over two. Multiplying the numerators gives us nine, and multiplying the denominators gives us four. Our final step is to multiply one-ninth by nine over four or nine-quarters. We could multiply one by nine and nine by four. However, we might notice at this point that we can cross-cancel or cross-simplify first.
As there is a nine on the top and bottom, we can divide both of these by nine. One multiplied by one is equal to one. And one multiplied by four equals four. The calculation negative one-third squared multiplied by three-halves squared is equal to one-quarter or one-fourth.
We will now look at an alternative method to solve this problem. We know that negative one-third squared is equal to negative one-third multiplied by negative one-third. Three over two or three-halves squared is equal to three over two multiplied by three over two. We were asked to multiply these two terms, giving us negative one-third multiplied by negative one-third multiplied by three-halves multiplied by three-halves.
At this stage, we can cross-simplify or cross-cancel. There is a three on the top and on the bottom. There is another three on the top and bottom, which also cancel. On the top, we’re now left with negative one multiplied by negative one multiplied by one multiplied by one. Multiplying two negatives gives us a positive. Therefore, negative one multiplied by negative one is one. On the bottom, we have one multiplied by one multiplied by two multiplied by two. This gives us four. Therefore, once again, the answer is one-quarter.