Solve each equation. See Examples 8 and 9. (x+5)^2/3+(x+5)^1/3-2...

Question

Solve each equation. See Examples 8 and 9.  (x+5)^2/3+(x+5)^1/3-20=0

Accepted Answer

Hey, everyone here we are asked to solve for X in the following equation where we have the quantity of X plus seven raised to the power of 2/ plus the quantity of X plus seven raised to the power of 1/5 minus six is equal to zero. Now, normally we would want to factor the left side of this equation to solve for X. But we see that it would be rather complicated to do so since we have X as a quantity raised to a power of a fraction, so we want to begin by simplifying our given equation and we can do this by letting the variable U the equal to the quantity of X plus seven raised to the power of 1/5. And now, with this statement here, we can now rewrite our given equation in a much simpler form. So our first term just becomes U squared and then we just have plus you And then our constant remains the same as -6 and this is all still equal to zero. So now looking at our rewritten equation, we see that this will be much easier to start factoring. So we can go ahead and start factoring by focusing first on our middle term here. Well, we want this positive view to be split into two terms whose sums to pas view, but the product of their coefficients must be equal to negative six. So doing this, our equation becomes U squared plus three U minus two, U minus six is equal to zero. So now we just want to check our middle terms here that we put into our equation. So we want to check that their sum is equal to positive you. So three U minus to you does in fact simplify to just positive you. And then we want to check their coefficients. So we have three times negative two, which does give us negative six. So these two terms work. So now we can move on to factoring this expression here. And we want to begin by looking at our first two terms and we see that they have a common factor of you so we can factor out you and it gives us you times the quantity of you plus three. And now looking at our last two terms, they have a common factor of negative two. So factoring that out, we get minus two times the quantity of you plus three. And this expression is is still equal to zero. And now we see that both terms have a common factor of you plus three. So we can go ahead and factor out this expression. And now our equation becomes the quantity of you plus three times the quantity of U minus two is equal to zero. So now we just need to recall the zero property, the zero factor property which tells us that we can set both individual expressions equal to zero. So we can start with the first part of our expression, which is you plus three. And again, we can set this equal to zero. So now we want to solve for you. So we first just subtract three from both sides. And we see that one of our values for you is negative three. And now we want to repeat this with the other part of our expression which is U -2. And we set this equal to zero. And now again, we need to solve for you. So we just need to add two to both sides. And we see that the other value we get for you is positive two. So now originally we are asked to solve for X. So our next step is to just substitute back in our original expression that we set equal to you for each value that we just found for you. So doing this, we can start first with our negative three equation. So we have the quantity of X plus seven raised to the power of 1/5 is equal to negative three. And now to solve for X, we first need to get rid of the power here. Which is 1/5. And to do this, we need to recall the power to a power rule for exponents which tells us that A to the power of N raised to the power of M is equivalent to A, to the power of N times M. So with this property in mind, we need to raise the expression on the left hand side to the power of five. So that way we can get rid of the 1/5 exponents. And because we raised the left side to the power of five, we also need to repeat this for the right hand side. And so now simplifying, we get X plus seven is equal to. Now the quantity of negative three raised to the fifth power simplifies to negative 243. And now just finishing up solving for X, we need to subtract seven from both sides. And we see that one of our values for X is just negative 250. So now we just need to repeat this with the other value we got for you, which was two. So we will have the quantity of X plus seven raised to the 1/5 power is equal to positive two. And now we just repeat what we just did earlier and we raise both sides to the power of five. And now simplifying, we get X plus seven is equal to, to raise to the power of five, which simplifies to just 32. And now we subtract seven from both sides to solve for X. And we see that X is equal to 25. So positive 25 is another solution for X. So with our two Values that we found for X, we can rewrite these solutions as a solution set. So we have the solution set of -250 as well as positive 25. So with these two solutions, we are left with answer choice d thanks for watching. I hope you found this video helpful and I'll see you again next time.

Solve each equation. See Examples 8 and 9. (x+5)^2/3+(x+5)^1/3-20=0

Key Concepts

Exponents and Radicals

Substitution Method

Polynomial Equations

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