Question Video: Simplifying Algebraic Fractions Mathematics • 10th Grade

Find the quotient of (26𝑎⁷𝑏⁵ − 9𝑎³𝑏⁵)/𝑎²𝑏.

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Video Transcript

Find the quotient of 26 𝑎 to the power of seven 𝑏 to the power of five minus nine 𝑎 cubed 𝑏 to the power of five all over 𝑎 squared 𝑏.

In order to actually solve this problem, what I wanna do is I actually want to break it into two parts. And we can actually do it using this relationship here, that says that 𝑎 plus 𝑏 over 𝑐 is equal to 𝑎 over 𝑐 plus 𝑏 over 𝑐.

And we can actually use that to split our question into two parts. Because we can actually say that our expression is equal to 26 𝑎 to the power of seven 𝑏 to the power of five over 𝑎 squared 𝑏 plus negative nine 𝑎 cubed 𝑏 to the power of five over 𝑎 squared 𝑏. Okay, great!

So now we’ve got this, we’re actually going to simplify in two parts. And to enable us to actually simplify these different parts I’ve split it into. We’re actually gonna use one of our exponent rules. And this rule states that 𝑎 to the power of 𝑚 divided by 𝑎 to the power of 𝑛 is equal to 𝑎 to the power of 𝑚 minus 𝑛.

So if we’re actually dividing two terms, where they’re the same term, then actually what we can do is subtract the powers. So let’s go ahead and use this over here. So first of all, I’m gonna have 26 and then 𝑎 to the power of seven minus two because seven would be like our 𝑚 and two would be like our 𝑛 in our exponent rule. And then we get 𝑏 to the power of five minus one.

And that’s, again, because our five is like our 𝑚 and then our one will be like our 𝑛. And that’s just remembering that when it’s just 𝑏 on its own, it’s like 𝑏 to the power of one. So great! And now we can actually fully simplify the left-hand side, which leaves us with 26 𝑎 to the power of five 𝑏 to the power of four. Great! So now let’s get on to the right-hand side.

So first of all, we’re gonna get negative nine 𝑎 to the power of three minus two. And then we’ve got 𝑏 to the power of five minus one. So great! Now we can fully simplify this. So that means our right-hand side, fully simplified, is negative nine 𝑎𝑏 to the power of four. So great! Final thing we need now to know is to bring it all together.

So it says this is all equal to 26 𝑎 to the power of five 𝑏 to the power of four plus negative nine 𝑎𝑏 to the power of four. Then we finally tidy it up, and we can say that our final answer is that the quotient of 26 𝑎 to the power of seven 𝑏 to the power of five minus nine 𝑎 cubed 𝑏 to the power of five over 𝑎 squared 𝑏 is equal to 26 𝑎 to the power of five 𝑏 to the power of four minus nine 𝑎𝑏 to the power of four.