Question Video: Simplifying Rational Algebraic Expressions Using Laws of Exponents Mathematics • 9th Grade

Video Transcript

Simplify 𝑎 raised to the power two over five multiplied by 𝑎 to the power of a half to the power five all over 𝑎 to the power three over two multiplied by 32𝑎 to the power eight to the power one over five, where 𝑎 is a positive constant and 𝑎 does not equal one.

We’re given an exponential expression where we have the positive base 𝑎, which is not equal to positive one, to various powers or exponents within a fraction. And so we have various exponential expressions within a rational expression. To simplify this, we’ll use the rules or laws of exponents. So let’s make a note of these before we start.

First, we have the product rule, where we saw our exponents. Second, we have the power rule, where we multiply our exponents. Our third rule or law is the power of products. Number four is the negative exponent law. And number five is the law for rational or fractional exponents. And in general, 𝑎 and 𝑏 are nonzero integers, where in our case, as noted, 𝑎 is not equal to one. So let’s see now how we can use these laws to simplify the given expression.

Let’s consider first the numerator. We have 𝑎 raised to the power two over five multiplied by 𝑎 to the power of a half all to the power five. Using our second law, that’s the power rule, on the second term, 𝑎 to the power of a half all to the power five, we simply multiply our exponents. That is one over two multiplied by five, which is five over two, so that our numerator is 𝑎 to the power two over five multiplied by 𝑎 to the power five over two.

Now we can apply the first law, that is, the product law. Where given a product of exponential expressions with the same base, we add the powers. In our case, that’s two over five plus five over two, which is 29 over 10. Our numerator is then 𝑎 raised to the power 29 over 10.

So now let’s look at our denominator. We have 𝑎 raised to the power three over two multiplied by 32𝑎 to the power eight all to the power one over five. And for our second term here, we can use rule number three, that is, the rule for a power of a product. And so our denominator is 𝑎 to the power three over two multiplied by 32 to the power one over five multiplied by 𝑎 to the power eight over five, where we’ve used rule number two for 𝑎 to the power eight all to the power one over five. So we’ve multiplied eight by one over five.

The middle term here has a numerical base. And we can simplify this using rule number five for rational exponents. If we recall that two to the power five is 32, then two is equal to 32 raised to the power one over five, which is the fifth root of 32. So far then, our denominator is 𝑎 raised to the power three over two multiplied by two multiplied by 𝑎 to the power eight over five.

And so now if we bring the constant two to the front, we see that we’re going to be able to use rule number one to simplify our denominator further. And we do this by summing our powers of 𝑎. So now we have two multiplied by 𝑎 raised to the power three over two plus eight over five. Three over two plus eight over five is 31 over 10. And so our denominator is two multiplied by 𝑎 raised to the power 31 over 10.

So now making some space, our expression is now simplified to 𝑎 to the power 29 over 10 divided by two 𝑎 to the power 31 over 10. And now splitting this into a product of fractions, we can use rule number four on one over 𝑎 to the 31 over 10 so that one over 𝑎 to the 31 over 10 is 𝑎 to the negative 31 over 10. So now we have 𝑎 to the 29 over 10 divided by two multiplied by 𝑎 to the negative 31 over 10.

So, finally, we can again use our product rule number one, where we sum our two powers of 𝑎. And 29 over 10 plus negative 31 over 10 is negative two over 10. And that’s negative one over five. And we have 𝑎 raised to the power negative one over five over two. So the given expression simplifies to 𝑎 raised to the power negative one over five over two.

We note that this could also be written one over two 𝑎 to the power one over five. That’s using rule number four. And this could also be written as one over two times the fifth root of 𝑎 using rule number five. However, we leave our answer in the form 𝑎 to the power negative one over five divided by two.