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
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
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
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.