Video: Simplifying Algebraic Fractions Using Properties of Exponents

Video Transcript

Simplify 𝑥 to the 𝑛 plus nine 𝑦 to the 𝑛 plus four over 𝑥 to the 𝑛 plus three times 𝑦 to the 𝑛.

Here we notice that we have 𝑥 and 𝑦 in both the numerator and the denominator. Our first step to simplifying will be trying to get all of the 𝑥 and 𝑦s in the numerator. Remember that you can move a base to a certain exponent and the denominator to the numerator by taking its negative exponent.

𝑥 to the 𝑛 plus nine was already there. And if we take 𝑥 to the negative 𝑛 plus three, we can bring that value to the numerator. 𝑦 to the 𝑛 plus four was already in the numerator. And we want to bring 𝑦 to the 𝑛 into the numerator. So we take its negative exponent, 𝑦 to the negative 𝑛.

What is the mathematical operation happening here? It’s multiplication. All of these exponents are being multiplied together. And that means we need the exponent product rule. It tells us that 𝑥 to the 𝑎 power times 𝑥 to the 𝑏 power equals 𝑥 to the 𝑎 plus 𝑏. When your bases are the same and you’re multiplying these two exponents together, you do that by adding the two exponents.

We have two exponents with the base of 𝑥. We keep the base of 𝑥, and then we add their exponents together, 𝑛 plus nine. Be careful with your negative there. You need to distribute that negative value across the 𝑛 and the three. When we do that, we get 𝑛 plus nine minus 𝑛 minus three.

Combine like terms. Plus 𝑛 minus 𝑛 cancels out. Nine minus three equals six. 𝑥 should be taken to the sixth power. We also have two exponents with a 𝑦 as a base. We’ll take 𝑦 to the 𝑛 plus four plus negative 𝑛, 𝑛 plus four minus 𝑛. Combining like terms, 𝑛 minus 𝑛 cancels out, leaving us with four. 𝑦 is being taken to the fourth power.

There’s one more important thing. Remember that we were multiplying the 𝑥s and the 𝑦s together, and that means the final simplification is 𝑥 to the sixth times 𝑦 to the fourth.