Given that 𝑛 one of 𝑥 equals 𝑥 plus 16 over eight minus 𝑥, 𝑛 two of 𝑥 equals nine 𝑥 plus 144 over eight minus 𝑥, and 𝑛 of 𝑥 equals 𝑛 one 𝑥 divided by 𝑛 two 𝑥, determine 𝑛 of 𝑥 in its simplest form.
We’re going to be working out 𝑛 of 𝑥, which is the quotient of 𝑛 one of 𝑥 and 𝑛 two of 𝑥. So let’s divide 𝑥 plus 16 over 𝑥 minus eight — remember that’s 𝑛 one — by 𝑛 two, which is known as 𝑥 plus 144 over 𝑥 minus eight. Before we do that, we’re going to need to simplify the expressions in each of these set of parentheses. By adding a denominator of one to 𝑥 and nine 𝑥, we see that we can add the fractions in each expression. We’ll create a common denominator in this first expression by multiplying the first term by 𝑥 minus eight over 𝑥 minus eight. That gives us 𝑥 times 𝑥 minus eight over 𝑥 minus eight plus 16 over 𝑥 minus eight. We will multiply nine 𝑥 over one by 𝑥 minus eight over 𝑥 minus eight too. That gives us nine 𝑥 times 𝑥 minus eight over 𝑥 minus eight plus 144 over 𝑥 minus eight.
And then, we add the numerators. Our first expression becomes 𝑥 times 𝑥 minus eight plus 16 all over 𝑥 minus eight. And our second expression becomes nine 𝑥 times 𝑥 minus eight plus 144 over 𝑥 minus eight. Then, we recall that to divide by a fraction, we multiply by the reciprocal of that same fraction. In this case, we multiply by 𝑥 minus eight over nine 𝑥 times 𝑥 minus eight plus 144. And then if we look carefully, we see we can cross cancel. We can divide by 𝑥 minus eight. We then multiply the numerator of the first fraction by the numerator of the second and repeat that process for the denominator. So we get 𝑥 times 𝑥 minus eight plus 16 times one, which is just that our original expression, over nine 𝑥 times 𝑥 minus eight plus 144.
Next, we distribute our parentheses. We multiply 𝑥 by 𝑥 and 𝑥 by negative eight. And we get 𝑥 squared minus eight 𝑥. Similarly, we multiply nine 𝑥 by 𝑥 and nine 𝑥 by negative eight. So we end up with 𝑥 squared minus eight 𝑥 plus 16 over nine 𝑥 squared minus 72𝑥 plus 144. And in fact, it might look like we’re finished. But we need to find 𝑛 of 𝑥 in its simplest form. So here, we notice that we have a common factor in the denominator. That common factor is nine. And if we factor nine out of the denominator, we get nine times 𝑥 squared minus eight 𝑥 plus 16. Now, notice we have a common factor. We can divide both the numerator and the denominator by 𝑥 squared minus eight 𝑥 plus 16. And so, we’re left with one-ninth.
𝑛 of 𝑥 equals one-ninth.