Question Video: Evaluating Algebraic Fractions with Negative Exponents Involving the Order of Operations Mathematics • 6th Grade

Given that 𝑥 = 6 + 3 × 8 ÷ 4 and 𝑦 = 4 + (8 × 2) − 3², evaluate (𝑥/𝑦)⁻².

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Video Transcript

Given that 𝑥 equals six plus three times eight divided by four and that 𝑦 equals four plus eight times two minus three squared, evaluate 𝑥 over 𝑦 to the negative two.

First, we’ll have to solve for our two variables. And then, we’ll plug in the information we find into a third equation. When it comes to solving for 𝑥 and 𝑦, we need to carefully follow the order of operations.

Starting on the left with our equation 𝑥 is equal to, there are no parentheses, there are no exponents. So next, we move to multiplication and division in order from left to right. In order from left to right, the first multiplication or division we get to is three times eight equals 24. And every other operation will just bring down exactly as it’s written.

We haven’t finished our multiplication or division step. So we take 24 and we divide it by four, which equals six. Bring down the rest of the operations. We’re in the addition and subtraction stage. Six plus six equals 12. Our 𝑥-value is equal to 12.

Now, we move on to solve for 𝑦. But we’ll again follow the order of operations. Solving for anything in parentheses first, eight times two equals 16. Copy the rest of the operations down exactly as they’re written. Now, we’re finished with parentheses and we can move on to exponents. We have three squared equals nine.

We copy down the other two operations, move on to the multiplication and division step. There are no multiplication or division operations. So we add and subtract from left to right four plus 16 equals 20. Bring down the rest of the problem. 20 minus nine equals 11. Our 𝑦-value equals 11.

Now, we have to take these data for 𝑥 and 𝑦 and plug them in: 12 over 11 to the negative two power. What we want to do now is distribute our negative two power: 12 to the negative two over 11 to the negative two. And here is where we want to be very careful. 12 to the negative two is equal to one over 12 squared. Now, we have one over 12 squared times one over 11 to the negative two. But one over 11 to the negative two is equal to 11 squared over one.

If you have a negative exponent in the denominator, you move it to the numerator and it becomes positive. Our new expression will look like this: one over 12 squared times 11 squared over one. Put them together and you have 11 squared over 12 squared. 11 squared is 121. 12 squared is 144.

Using the information we were given, 𝑥 over 𝑦 to the negative two is equal to 121 over 144.