Use the quotient rule to simplify the expressions in Exercises...

Question

Use the quotient rule to simplify the expressions in Exercises 23–32. Assume that x > 0. √150x^4/√3x

Accepted Answer

Hello, today we are going to be using the quotient rule of radicals to simplify the radical expression. Before we jump into this problem, let's quickly review what the quotient rule of radicals is. Let's say we have radical A divided by radical B. The quotient rule allows us to combine the division of two radicals into a single radical by taking the division of the inside arguments. So in other words, we can rewrite radical A divided by radical B as radical A divided by B, we're going to want to use this rule to simplify our given expression. Now, the expression given to us is the square root of 200 multiplied by Z to the eighth power divided by the radical five Z. So the first thing we can do is we can use the quotient rule to combine the division of two radicals into a single radical that will allow us to rewrite the expression as radical 200 divided by or multiplied by Z to the eighth divided by five Z. Now, what we can do is we can simplify the inside of the radical 200 is divisible by five by 40 times. So we can reduce 200 in the numerator to 40 reduce 5 to 1 Z to the eighth divided by Z will leave us with Z to the seventh. Because recall that if we take the division of two exponential values, all we are doing is taking the difference of the exponential values. So this will give us Z to the eighth minus one which will leave us with Z to the seventh in the numerator that will allow us to simplify the radical to be radical 40 multiplied by Z to the seventh. Now, what we want to do is you want to go ahead and simplify the inside argument further. Now recall that the radical allows us to simplify perfect squares into single terms. So what we want to do is you want to take 40 break it down into a product of simple terms and combine them into perfect squares. We will do the same with Z to the seventh. So what are two numbers that multiply together to give us 40? Well, 40 is the product of 10 and four. We can break down four further because it is a product of two multiplied by two and 10 is a product of five and two. So that means that we can rewrite 40 as two multiplied by two multiplied by two multiplied by five. Now keep in mind that we want to work with perfect squares. So if we group up the product of the 1st 22 terms, then we can rewrite the product of 40 as two squared multiplied by two, multiplied by five. And for now we're going to multiply five and two together and just leave it as 10. So the product of 40 is equal to two squared multiplied by 10. Now, if we take a look at the term Z to the seventh, if we want to break this down as a product of perfect squares, we can rewrite Z to the seventh as Z squared multiplied by Z squared multiplied by Z squared multiplied by Z. The reason why we can break down this Z to the seventh as this product is because if we multiply all these terms together, we will take the sum of all the exponents and the sum of all the exponents will give us seven. So using this breakdown of 40 this breakdown of Z to the seventh, we can rewrite the radical expression as the square root of two squared multiplied by 10, multiplied by Z squared multiplied by Z squared multiplied by Z squared multiplied by Z. Now, what we want to do is we want to remove every square root outside of the radical symbol. And we can do that by just writing the base of each square root, we have two squared which is a perfect squared and three iterations of Z squared. So we can simplify this radical to be two multiplied by Z multiplied by Z multiplied by Z multiplied by the square root of the terms remaining inside the radical, which is 10 times Z. So 10 Z finally, what we are going to do is we are going to simplify the coefficients in front of the square root symbol. The only thing that we can multiply together are the three Zs in the coefficient which will give us Z cubed. So that will leave us with two multiplied by Z cubed multiplied by the square root of 10 Z. There is no further way to simplify the radical expression. So this is going to be the simplest form of the original radical expression. And with that being said, the solution to this problem is going to be D. So I hope this video helps you in understanding how to simplify radical expressions using the quotient rule. And I'll go ahead and see you all in the next video.

Use the quotient rule to simplify the expressions in Exercises 23–32. Assume that x > 0. √150x^4/√3x

Key Concepts

Quotient Rule

Radical Expressions

Simplification of Algebraic Expressions

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