Which of the following is equal to two and three-quarters squared? Is it (A) seven and nine sixteenths, (B) four and nine sixteenths, (C) five and a
half, (D) two and nine sixteenths, or (E) twenty-two eighths?
Our first step is to turn the mixed number two and three-quarters into an improper or
top-heavy fraction. Two is equal to eight-quarters as there are four quarters in one whole. This means that two and three quarters is the same as eight-quarters plus
three-quarters. When adding two fractions with the same denominator, we just add the numerators. Therefore, eight quarters plus three quarters is equal to eleven-quarters.
A quicker way to change a mixed number into an improper fraction is to multiply the
whole number, in this case two, by the denominator four and then add the
numerator. Two multiplied by four plus three is equal to 11. Therefore, two and three-quarters is equal to eleven-quarters. We can therefore write two and three-quarters squared as eleven-quarters squared. Any fraction 𝑎 over 𝑏 that is raised to the power of 𝑥 can be rewritten as 𝑎 to
the power of 𝑥 divided by 𝑏 to the power of 𝑥. This means that 11 over four or eleven-quarters squared is the same as 11 squared
divided by four squared.
11 squared is equal to 121, and four squared is equal to 16. Therefore, 11 squared divided by four squared is equal to 121 over 16. We now need to turn this top-heavy or improper fraction back into a mixed number. In order to do this, our first step is to work out how many 16s go into 121. Six times 16 is equal to 96. Seven times 16 is equal to 112. And eight times 16 is equal to 128. We can therefore say that 16 divides into 121 seven times with a remainder of
nine. As 121 minus 112 is equal to nine, the improper fraction 121 over 16 can be rewritten
as seven and nine sixteenths. This means that the correct answer was A: seven and nine sixteenths is equal to two
and three-quarters squared.