Perform the operation and simplify. Assume all variables represent non negative real numbers. 5

Perform the operation and simplify. Assume all variables...

Key Concepts

Radical Notation and Fractional Exponents

This concept involves rewriting radical expressions, such as nth roots, as expressions with fractional exponents. For example, the fourth root of a variable raised to a power can be written by dividing the exponent by 4. This technique is fundamental because it allows use of exponent rules in subsequent steps.

Properties of Exponents

This concept encompasses the rules for manipulating expressions with exponents, such as multiplying expressions with the same base by adding the exponents and power raising power rules. These properties are crucial for simplifying expressions after converting any radicals to fractional exponents.

Combining Like Terms

After rewriting expressions in terms of exponents, like terms—those with the same base and exponent—can be added together by combining their coefficients. This step uses the distributive property and is key to simplifying algebraic expressions.

Transcript

00:01 All right, so our goal here is to combine these two terms and basically reduce this expression into its most simplest form.

00:09 Now, we're taking the fourth root in both of our radicals, but the coefficients are the actual radicans underneath our radicals are not matching.

00:20 So that means we need to actually simplify these in order to actually be able to combine them.

00:28 And so what we're going to want to do here is go ahead and expand out the factors of our radicals so that we can see what we can actually take the fourth root up.

00:41 So a to the seventh is the same thing as a to the fourth power times a to the third power.

00:54 And so when you're multiplying these two, you'll just add four and three.

00:59 And that's how we get the seventh exponent.

01:02 And the reason why i did this was because a to the fourth power is something we can take the fourth root of very easily.

01:10 And so that will allow us to actually remove that factor from underneath the radical.

01:15 And so when we look at our radical here, fourth root of a to the fourth times a to the third, i will be able to take the fourth root of a to the fourth, which means i can remove it from underneath the radical, and put one of the factors out front.

01:48 And that will just be multiplied by the rest of the coefficients in front of this particular radical.

01:58 So in this case, 5a.

02:02 Now, let's go ahead and look at the second half of our expression.

02:08 So we have a plus sign down here and we know we have the fourth root.

02:15 And we're going to be doing the same thing of expanding it out.

02:18 So let's go ahead and look at the factors that we can get for a to the 11th.

02:22 And again, our goal here is wanting to take the fourth route.

02:25 So we're going to want to pull out a to the fourth terms as much as possible.

02:31 So i can do a to the fourth times a to the fourth.

02:39 That'll get me a to the eighth.

02:42 So i still have three extra a.

02:46 So a to the third would be remaining...