in-3-41-express-each-product-in-simplest-form-variables-in-the-radicand-with-an-even-index-are-no-23

Question

In $3-41$ , express each product in simplest form. Variables in the radicand with an even index are non-negative.
$$
\sqrt{2}(2+\sqrt{2})
$$

Sarah Wharton · Accepted Answer

in this question, we&#x27;re solving for the square root of two times the quantity of two plus route to. To do so will be using the distributive property showing here where we distribute this route to OutFront to both of the terms. Inside the parentheses, we&#x27;ll get route to times two, plus a route to times route to this will give us to root. Two rewritten toe have the whole number Elton front, plus the square root of two times to. Because any two routes multiplied together, we can just multiply them in one big route, so this will be equal to to root two plus the square root of four, which is just equal to to root two plus two.

M S · Answer

so we have to queue. Brutal, too. Time security. Four. So because they have the same radical, we can combine everything under the radical so you can have two times for and two times four is eight, and the Q crew debate is too. Cube root works just like the square root. So square root is like two times two to give with sport. But the cube root is two times two times two or to give us eight. That&#x27;s how that works.

Sarah Wharton · Answer

in this problem we&#x27;re solving for the square root of a over three times the square root of a squared over four. And for this problem, I&#x27;m gonna choose to simplify this term on the right first, because I noticed that both the top and the bottom are perfect squares. So this leaves me with the square root of a over three times the square root of a squared who were square root of four. This is equal to the square roots of a over three times A over to now in to deal with this term on the right. To do that, I&#x27;m going to similarly separate it into a top and bottom. So and I&#x27;m gonna put this term out in front just because I&#x27;m done dealing with it. So this is equal to a over two times the square root of a over the square root of three. Now, to get rid of this three on the bottom, when you do is multiply this term in particular by Route three over route three. Now, Route three over Route three is just a form of one. So we&#x27;re not fundamentally changing the equation at all were simply multiplying by one. But what that does is that kids get gets rid of the radical on the bottom here with those route threes. So this is going to be equal to a over to times Route three A. From combining the three and a there divided by route three times, Route three, which is just three now if I bring this on to another line and multiply altogether, this gives me a Times Route three a divided by two times three. So this is equal to a over six times Route three A and that is our final answer.

M S · Answer

you have to square It affects summer to Times Square. That expired over too. So the first thing I want to do is I&#x27;m going to split apart the radicals. So we have a spirit of X over the spirit to time screw of X squared over this swearing too. So our talk this square Uranus squared were canceled. Still have it times the spirit of it. Oh, we&#x27;re our denominator and a spirit of two times greater too will give us two. And that&#x27;s a CE faras. We can simplify this expression.

M S · Answer

so we have to square 2 12 squared. This one is really simple because square root and squared are opposites of each other. This very well. Just cancel out and we&#x27;re left with Swell. Another way to prove this is that the square to 12 Times Square to 12 and the square roots also cancel out. So it was Lawrence&#x27;s.

Sarah Wharton · Answer

in this question. We&#x27;re solving a four to minus root five times two plus Route five squared. And the first thing we want to do is expand this expression here. The two plus Route five squared to do so I&#x27;m gonna rewrite it as two plus route five times two plus Route five. And we&#x27;ll use foil first outside inside. Last as a way to expand this and do the multiplication. So first we have the numbers out in front. That&#x27;s two and two. So two times two is equal to four. We&#x27;re gonna do the outside numbers next to Times Route five. So it&#x27;s plus to Route five. Inside comes next. So two times, Route five again is to route five. And then finally, the last numbers Route five, Andrew five, route five times. Route five is just going to be five because any number inside a radical times itself, the same number and the radical is just gonna be the original number. So do plus five here and now we need to combine like terms because both of these two terms have the Route five. We can consider them, um, similar terms. So we&#x27;re gonna add together the four and the five combining like terms to get nine. And then we&#x27;ll do the two. Route five plus two, Route five. That would be four root five. So we have nine plus four root five. This now replaces this term right here in our original equation. So gonna rewrite this as Tu minus route five times nine plus four root five. And now we can do foil again. First outside, inside last. So first we get two times minus 18. We&#x27;re gonna do the outside next. That&#x27;s two and Route five. Our started to end four root five. So that would be eight. Route five. Since we multiply the whole numbers together inside comes next. That would be negative. Route five times nine. So this will be minus nine. Route five, then finally, we have negative route five times four root five. And in this case, we&#x27;re going to multiply the roots. Together, we would have a negative four times route five times, five times five. So this would be minus four times five or minus 20. Now, we&#x27;re going to combine like, terms again. So I&#x27;m just gonna make some space down here for us to do that When combining like terms. We notice that this is 85 negative nine, Route five so they could be treated as, like terms. So I&#x27;m gonna rewrite this just tohave little bit more clarity this equal to 18 plus eight. Route five minus nine. Route five minus four times five is 20. Our first set of like terms is 18 and negative. 20 18 minus 20 is negative two. And our second set of like terms is positive. Eight. Route five and Negative Minor. Five eight minus nine. Route five is negative. One. Route five. And since we have a coefficient of one, our most simple way of writing this be negative two minus Route five, which is our final answer.

Problem

In $3-41$ , express each product in simplest form…

Problem 25 Medium Difficulty

In $3-41$ , express each product in simplest form. Variables in the radicand with an even index are non-negative.
$$
\sqrt{2}(2+\sqrt{2})
$$

Answer

$2 \sqrt{2}+2$

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$2 \sqrt{2}+2$

Algebra 2 and Trigonometry

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