Find the value of four-thirds squared divided by negative eight-ninths squared, giving your answer in simplest form.
Our first step is to square each of the fractions in the parentheses or brackets. Four-thirds squared is equal to four-thirds multiplied by four- thirds. When multiplying fractions, we multiply the numerators and separately multiply the denominators. Four multiplied by four is equal to 16, and three multiplied by three is equal to nine. Therefore, four-thirds squared is equal to 16 ninths or 16 over nine. We need to carry out the same process to square negative eight-ninths. We need to multiply negative eight-ninths by negative eight-ninths. Multiplying two negative numbers gives us a positive answer. Eight multiplied by eight is equal to 64, and nine multiplied by nine equals 81. This means that negative eight-ninths squared is equal to 64 over 81.
The original question can therefore be rewritten as 16 over nine divided by 64 over 81. When dividing two fractions, you might have heard the acronym KCF: we keep the first fraction the same, we change the sign from a division sign to a multiplication sign, and we flip the second fraction. 64 over 81 becomes 81 over 64. Dividing by any fraction is the same as multiplying by its reciprocal. At this stage, we could multiply 16 by 81 and nine by 64 and then simplify our answer. However, if possible, it is easier to simplify or cancel. At this stage, we can divide the nine on the bottom of the first fraction and the 81 on the top of the second fraction both by nine. Nine divided by nine is equal to one, and 81 divided by nine is equal to nine. In the same way, we can divide the top or numerator of the first fraction and the bottom or denominator of the second fraction both by 16. 16 divided by 16 is equal to one, and 64 divided by 16 is equal to four.
We are left with one over one multiplied by nine over four. This is equal to nine over four or nine quarters. We could write this improper or top-heavy fraction as a mixed fraction. Nine over four or nine-quarters is equal to two and one- quarter as nine divided by four is equal to two remainder one. This means that the value of four-thirds squared divided by negative eight-ninths squared in its simplest form is nine- quarters or two and a quarter.