Given that 𝑎 equals six-fifths and 𝑏 equals negative twenty-four fifths, find the absolute value of 𝑎 squared divided by 𝑏 squared.
First, we need to replace 𝑎 with six-fifths and 𝑏 with negative twenty-four fifths. We need to follow the order of operations, which means we need to take care of the exponents before we divide. So we need to square six-fifths and square negative twenty-four fifths. When you square six-fifths, you’re squaring six and you’re squaring five. Same thing with negative twenty-four fifths, you’re squaring negative twenty-four and you’re also squaring five. Six squared is thirty-six. Five squared is twenty-five. Now negative twenty-four squared, that’s equal to five hundred and seventy-six. Because when you square negative twenty-four, that’s negative twenty-four times negative twenty-four, and two negatives multiply to be a positive. So again, negative twenty-four squared is equal to positive five hundred and seventy-six. And then five squared is equal to twenty-five. Now in order to divide fractions, we actually change the division sign to a multiplication sign and flip the second fraction. We’re essentially multiplying by this second fraction’s reciprocal.
So again, we changed the division sign to a multiplication sign and we flipped the second fraction. So now it’s twenty-five over five hundred and seventy-six. When multiplying fractions, we can reduce anything on the numerator with anything on the denominators. So this means we can go ahead and simplify the twenty-fives. They essentially cancel. Twenty-five goes into twenty-five once. We can also simplify the thirty-six and the five hundred and seventy-six because thirty-six goes into thirty-six once, and thirty-six goes into five hundred and seventy-six sixteen times. So on the numerators, we have one times one, which is equal to one. And then on denominators, we have one times sixteen, which is sixteen. And now the absolute value sign will turn any number positive. So our positive one sixteenth will stay positive one sixteenth.
So given that 𝑎 was equal to six-fifths and 𝑏 was equal to negative twenty-four fifths, the value of the absolute value of 𝑎 squared divided by 𝑏 squared is equal to one sixteenth.