Perform the indicated operations. Assume all variables represent ...

Question

Perform the indicated operations. Assume all variables represent positive real numbers. (3√2 + √3) (2√3 - √2)

Accepted Answer

Welcome back. I am so glad you're here. We are given this expression parentheses 11 square root seven plus the square root of 11. And parentheses times a new parentheses seven square root 11 minus the square root of seven. We are asked to simplify this and evaluate using exact values. Alright, so we are going to be using the foil method to multiply here. Where we're going to take the firsts from each set of parentheses, multiply those by each other and then add that to the Outsides of each set of parentheses, multiplied by each other and add that to the insides of each set of parentheses, multiply those by each other and add that to the lasts of each set of parentheses, multiplied by each other. So let's write that out. Our firsts are 11 squared of seven times seven square root. 11 Plus. Our outsides multiplied together. That's 11 square root. Seven times the negative square root of seven Plus. Our insides multiplied together. That is the square root of 11 times seven square root of 11. And add that to our lasts multiplied together. Which is the square root of 11 times negative square root of seven. That is so many sevens and elevens. All right now let's do that multiplication. So our first we're going to take the terms in front of the radicals and multiply those by each other. So 11 times seven. That's 77. And then we'll take the terms inside the radicals and multiply those by each other. So inside the radicals we have seven times 11 which will give us the square root of Plus. Now let's do the outsides. Outside of the radicals, we have 11 times negative one. So that will be -11. And inside the radicals we have seven times seven, so that's the square root of 49. Now, for our insides in front of the radicals we have one time seven, so plus seven. And inside the radicals we have 11 times 11. So we've got the square root of 121. Now for our lasts in front of the radicals we have one times negative ones will have a negative one or just minus. And then inside the radicals 11 times seven. So that's the square root of 77. Now we can simplify what's inside the square root of 49 and the square root of 121. So let's do that will bring down 77 square root 77-11 times and the square root of 49 is seven Plus seven times the square root of 121 is minus the square root of 77. And now we can do this multiplication here, we can take negative 11 times seven and we can take seven times 11. So I'll bring down 77 square root negative 11 times seven is minus positive seven times 11 is plus 77. Bring down that minus square root of 77. Now we can combine like terms are like terms here are the ones with the square root 77s and then the ones with no radicals, this minus 77 plus 77. So starting with the terms that have radicals, 77 square root 77 minus this is like a minus one square root 77. So it's like we've got 77 of these square root 70 seven's but someone took one away and now we only have 76 square root 70 seven's left. Now combining our terms without the radicals negative plus 77. Will those cancel out? That's just zero. So our final answer is 76 square root 77. We look at our answer choices and that matches with answer choice. A well done. We'll catch you on the next one.

Perform the indicated operations. Assume all variables represent positive real numbers. (3√2 + √3) (2√3 - √2)

Key Concepts

Radical Expressions

Distributive Property

Combining Like Terms

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