### Video Transcript

Evaluate 0.8 repeating divided by the absolute value of negative five-fourths, giving the answer in its simplest form.

We want to evaluate this expression. To do that, we need to rewrite this recurring decimal as a fraction. 0.8 repeating is eight-ninths written as a fraction. Generally, we just memorize how to translate recurring decimals into fractions. But if you forget, you could think of it like this. If 𝑥 equals 0.8 repeating and we multiply both sides of our equation by 10, we get 10𝑥 equals 8.8 repeating. And then, if we subtract 𝑥 from both sides, 10𝑥 minus 𝑥 equals nine 𝑥. And because we know that 𝑥 equals 0.8 repeating, we can subtract 0.8 repeating from the right-hand side so that our new statement says nine 𝑥 equals eight. And then we divide both sides of the equation by nine, which means that 𝑥 equals eight-ninths.

Okay, back to the problem at hand. We’ve rewritten 0.8 repeating as eight-ninths, and now we need to deal with this absolute value. We have the absolute value of negative five-fourths. We know that the absolute value is the distance from zero. And that means we need the nonnegative value of what’s inside the absolute value bars. In this case, instead of negative five-fourths, we would have positive five-fourths so that our new expression says eight-ninths divided by five-fourths.

But when we’re dealing with division and fractions, we change the division to multiplication and we multiply by the reciprocal. Instead of dividing by five-fourths, we’ll need to multiply by four-fifths. We now need to take eight-ninths, multiply it by four-fifths. It doesn’t seem like there’s anything we can simplify, so we multiply the numerators. Eight times four is 32. Nine times five is 45. And because 32 and 45 do not share any common factors apart from one, this is the simplest form.

0.8 repeating divided by the absolute value of negative five over four in its simplest form will be equal to 32 over 45.