in-11-38-evaluate-each-expression-in-the-set-of-real-numbers-sqrt3frac1000a3-a-neq-0

Question

In $11-38,$ evaluate each expression in the set of real numbers.
$$
\sqrt[3]{\frac{1,000}{a^{3}},} a 
eq 0
$$

Dilip Paruchuri · Accepted Answer

Hey, guys. So in this problem, we&#x27;re given the cube root of 1000 over a cube, and we&#x27;re also given the condition that are a is not equal to zero, which is important because our A&#x27;s in the denominator and this problem are variable A. So if it was equal to zero, this value would be undefined. So our parameter are given condition right here is just to make sure that our function is defined, that&#x27;s all. So we&#x27;re being asked to compute the real number real value expression for Cuba of 1000 over a tube. So let&#x27;s start that. So we have the original expression. Just gonna rewrite that over here. And so what I&#x27;m gonna do is I&#x27;m gonna split the cube root into the numerator and the denominator. So you have the cube root of 1000 in the numerator and the cube root of a cute and the denominator. Now, when when you&#x27;re dealing with roots, we want to take on cube roots. We want to get rid of the cube root, and the way we do that is if we take the cube root off something cute and unfortunately for us, our denominator is already written in a form of something Cubans were in as a cute, so cute brute of a cube is just a And now we want to write our numerator, which is 1000 as something cute. And luckily for us, 1000 is also a perfect cube. It&#x27;s equal to 10 cubed 10 times 10 is 100 100 times 10 is it 1000. So uh 1000 is equal to 10 cube Soren numerator is equal to uhm cube, root of 10 cute and key route of 10 Cube is the same thing is 10 So our numerator is equal to 10 and this is our answer right here. Our cube root of 1000 over a cute is just equal to 10 over a under the stipulation that are a is never equal to zero Because, like I said, if the AI was equal to zero, this value would be undefined. Thank you for listening, guys. And I hope this helps solve this problem

James Kiss · Answer

were asked to find the Q route of zero 0.0 01 Well, thank you, Root no clean 001 is the same thing as the Cube root. A 0.1 to third 0.1 of the third is 0.1 Now that we run, it is the perfect Q Bar Cube root and are perfect. You counter act there in verses of each other, and now this tells us that the cube root of 0.1 is simply zero 0.1.

Dilip Paruchuri · Answer

Hey, guys. So in this problem were given forth fruit of 0.256 and were asked to simplify this expression into a number or a value that&#x27;s in the real numbers. So the first step we see that we&#x27;re given 1/4 of route. The way we get rid of roots or fourth roots or any roots in general, is if we do the opposite. If we if we raise something to the X ray, something to that same power, meaning if you have the fourth route, if you take 1/4 root of something raised to the fourth power, the fourth route and that in the raising to the fourth Power cancels out and you&#x27;re just left with the number. So basically, what I mean is, if you&#x27;re taking the fourth, the root of let&#x27;s Say X to the fourth, the fourth route and the four and the four in the exploding cancel out and you&#x27;re just left with Simple X. And so we want to do the same thing in order to Ken throughout the fourth route, we want to write our 0.256 as something to the fourth power and threw some algebra. We see that points your tooth 56 is equal to 0.4 times itself four times. And I&#x27;m going to show you the algebra right here. So we have 0.4 times itself for times So 0.4 times zero point for is equal to 0.16 and 0.16 times 0.4 is equal to 0.64 And finally 0.64 times 0.4 is equal to zero 0.256 which is the value that we have in our expression. So what we could do is we could rewrite this expression as the fourth root of 0.4 to the fourth. Now, if you&#x27;ve been paying attention to squares and radicals, you all. You also can make the assumption that this can also be equal to the quantity negative zero point forward to the fourth as well. However, since when we&#x27;re taking when a problem gives us the sport, the route when it explicitly writes down a root function, we always take the principal root, meaning we always consider the positive value of the radical but just keep in mind that points year 256 is also equal to the quantity of negative 0.4 to the fourth as well. Just that just that in this problem, we only care about the positive Because we&#x27;re given the four through. That just implies that we take the principal route. So basically the fourth route and the, uh 0.4 to the fourth power, the fourth power and the four through it cancel out and you&#x27;re just left with 0.4 so forth. Fruit of 0.256 is equal to 0.4, and we just included are positive value. Just keep in mind that this answer could have been negative. But since we&#x27;re taking the principal route, we will. You just keep in mind that Ah, we have our positive value. Thanks for listening, guys. And I hope

Dilip Paruchuri · Answer

Hey, guys. So in this problem, we&#x27;re given the value were negative square root 0.4 and were asked to simplify this. And before we go on simplifying the expression fun fact factory ization or the basically multiplying Ah, little trick that&#x27;s gonna be important. The step is that 0.4 is equal to 0.2 times 0.2. Basically, it&#x27;s equal to 0.2 square. So important multiplication tip that we&#x27;re gonna use in this problem. So we know that negative square root 0.4 We can simplify that as native 0.2 squared. We know when you take the square root of something square, you&#x27;re just going to get that something. So this is gonna be negative of 0.2. So the value that you&#x27;re gonna get is native 0.2 and this is gonna be our answer to the simplify the expression problem of negative square root 0.4 This about this negative zero square is Europe was your four is equal to negative 0.2. Thank you for listening, guys. And I hope this helps

Dilip Paruchuri · Answer

All right. Hey, guys. So, in this problem, we&#x27;re taking the fifth root of negative one over 32. The problem asks us to simplify this expression, and it states that are answers should be something in the real play, the real numbers. And so let&#x27;s tackle this problem with them Algebraic steps, basically rewriting what we&#x27;re given first. So we&#x27;re given the fifth root of negative one over 32. Now, what we can do is we can do this next step where we basically split the fifth root apart into the numerator and into the denominator. So we have We&#x27;re taking the fifth root of the numerator, and then we&#x27;re taking the fifth root of the denominator. Now let&#x27;s take a look at the numerator first, the numerator. Well, in a previous problem, well, we already did a problem like this. We said our fifth root of negative one is basically the same as fifth root of negative one to the fifth. Because in that problem, we saw the negative ones of the fifth is equal to negative one. And when you take the fifth would have something to the fifth u get net. You get that something So when you take 1/5 root of negative one to the fifth u get negative want. So from this, we can easily see that our numerator evaluates to negative one. And now we take a look at her denominator. So, like that previous problem, we want to do a similar thing we want to find. We want to cancel our fifth root out, and the way we do that is, if we take the fifth root of something to the fifth. So we want to rewrite our 32 as some number two, the fifth Power. And luckily through algebra, um, we see that 32 is actually equal to two to the fifth. So you multiply two times itself five times and you get 32. And so when you do that, fifth root of two to the fifth is going to be equal 22 and two to the fifties 32. So we know that fifth root of 32 is just equal to two. So our answer is negative. 1/2 show negative. 1/2 is the answer to the radical expression. Ah, fifth root of negative of negative one over 32. Now just to, uh I&#x27;m just gonna say this verbally because there is another way we could have tackle this problem. We could have tackle this problem through a In this problem, we tackle the problem by splitting up, splitting our radical into the numerator and denominator. But what? But what we could have also done is we could have done the fifth root of negative one times the fifth root of one over 32. And in doing this, we could have just solved each of these individually and you get negative one times one have. And then you would still hopes you would still get your answer to be, uh, negative 1/2. Just if you used this way to solve the problem. What happened is you just need to recognize that one over 32 is just equal to 1/2 to the fifth power. Just say if you get the same ends or just a different process, different approach. Thanks for listening, guys. And I hope this

Dilip Paruchuri · Answer

Okay, guys. So in this problem, we&#x27;re given the fifth root of negative one, and we&#x27;re being asked to simplify this expression and in the problem of states that are expression or simplified answer is gonna be in is gonna be a real number. And so let&#x27;s start off. So as we&#x27;ve been tackling most of these problems, the first step to do is to rewrite our negative one in a way that our fifth root cancels out with, uh, with the negative ones with the river in form of our negative one. By that, let me explain that the way to get rid of Fifth Root how do you get rid of him through you take the fifth root of something to the fifth power. And in that way we can get rid of the fifth through, and we&#x27;re just left with that something in the middle. And in order to do the same thing with this problem, we need to rewrite our negative one as naked as something as basically as something So the fifth power. And in order to do that, we&#x27;re just gonna factor it out. We&#x27;re gonna see what mould, what? Five things multiplied by itself gives us negative one and thinking about that we get that negative one is actually just native one times itself five times. Now, why is that true? If you just think about it, we know that negative one times negative one is one. No, no, One times negative one is negative. One negative, one times negative One is one and one times negative. One is native simple calculations. And basically, from this, we could basically rewrite fifth root of negative one as the fifth root of negative one to the fifth power and going back to this step right here, the way we cancel out the fifth root is if we have taken the fifth period of something to the fifth Power. We just get that something. This case, we&#x27;re taking the fifth root of negative one to the fifth power, and as a result, we just getting negative one. So because of these steps, we can literally see that fifth root of negative one is just equal to negative one. Negative one is in the reels. So this satisfies the conditioner problem gave that our expression answer should be in the reels. And just to recap the key step that we made right here is that we connected. Negative one is where we basically made the, uh, algebraic step That negative one is equal to negative one to the fifth power. And that&#x27;s basically what allowed us to cancel out the fifth root and come down to our final expression. That fifth root of negative one is just equal to you. Negative one. Thank you for listening, guys. And I

Problem

In $11-38,$ evaluate each expression in the set o…

Problem 34 Hard Difficulty

In $11-38,$ evaluate each expression in the set of real numbers.
$$
\sqrt[3]{\frac{1,000}{a^{3}},} a \neq 0
$$

Answer

$\frac{10}{a}$

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$\frac{10}{a}$

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