Perform the operation and simplify. Assume all variables represent non negative real numbers. 16

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

Key Concepts

Combining Like Terms

After simplifying individual radical expressions, similar radical terms can often be combined into a single term through addition or subtraction. This process requires the expressions to have the same simplified radical part so that arithmetic operations can be applied directly.

Cube Root Properties

Cube roots have specific properties, such as the multiplicative property which states that the cube root of a product is the product of the cube roots. This property is vital for breaking down complex radicands into simpler, more manageable parts, particularly when perfect cubes are present.

Radical Expression Simplification

This concept involves rewriting expressions under a radical sign so that any factors that are perfect powers (such as perfect cubes for cube roots) are extracted. It allows you to write the radical in a simpler form by taking out these factors, thus reducing the complexity of the expression.

Properties of Exponents

Using the rules of exponents is fundamental when simplifying radical expressions, especially when variables are involved. The properties, such as rewriting a power as a product of smaller powers, help in identifying and extracting perfect power factors from under the radical.

Transcript

00:01 Okay, so in order to answer this question, we want to go ahead and make the variables underneath my radicals match.

00:12 And so what i want to do here is try to factor out anything that i can take the third root of so that i can remove it from my radical.

00:21 So if we go ahead and look at this first term here, 128, that is going to be the same thing.

00:32 As 2 times 64.

00:47 And 64 is going to be the same thing as 2 times 32.

00:57 32 is the same thing as 2 times 16, the same thing as 2 times 8, and 8 is the same thing as the same thing, as 2 times 2 times 2.

01:24 And so what i want to go ahead and do is rewrite my radical with all of these factors underneath.

01:35 So i got 1, 2, 3, 4, 5, 6, 7 2s.

01:43 And they're all going to be multiplied by each other.

02:01 And then i'll have my h squared added.

02:05 I can't factor this down any further to try to take the third root because it's only to the second power.

02:12 So it's going to have to stay under the radical.

02:14 But what i do want to do is find anything where i've got, anytime i have three of the same of something, i can go ahead and take the third root of it.

02:23 So here i've got a group of three twos.

02:28 I can go ahead and take the third route, remove them from the radical, and place one of the factors out front.

02:38 So i can go ahead and do this again with a second group of twos, and so this will just be two times two that i removed.

02:51 And i also need to remember to bring down my original coefficient of 16.

02:58 That just means everything here will be multiplied by 16 as well.

03:09 And then we also want to do a similar operation on my second half of this equation.

03:17 So 16 is going to be the same thing as 2 times 8, and 8 is the same thing as 2 times 2 times 2.

03:36 If we go ahead and rewrite this expression...

Please give Ace some feedback