Solve each equation. See Examples 8 and 9. x^-2/3+x^-1/3-6=0

Question

Accepted Answer

Hey, everyone here we are asked to solve for X in the following equation X to the power of negative 2/5 minus two X to the power of negative 1/5 minus three is equal to zero. Now, looking at our given equation, we want to factor the left side to solve for X. But since we have X raised to a negative fraction, we want to begin by simplifying our given equation so that it is easier to for factor. So we can do this by letting the variable U be equal to X to the power of negative 1/5. So now with this statement, we can rewrite our given equation as a simplified version where we will first have you squared to replace X to the power of negative 2/5. And then we will have minus two you to replace our second X term and then we will Have just -3 is equal to zero. So now looking at our rewritten equation, we see that the left side is much easier to start factoring. So we can go ahead and start factoring and we need to begin by taking our middle term here which is negative to you. And we need to split it into two terms which some to negative to you. But the products of their coefficients must be equal to negative three. So doing this, we will have you squared minus three U plus one, U minus three is equal to zero. So now we can just check our two new middle terms here. So summing these two terms, we have negative three you plus one U and this does in fact equal negative to you. So we're good In that case, but now we want to check their coefficients. So we have negative three times one, which does also equal -3. So we chose the correct two middle terms and we can move on to factoring this expression. So we can start with our first two terms in the left side of our equation. And we see they have a common factor of you. So we can factor out you and have you times the quantity of U -3. And then for our last two terms, we can factor out positive one. So we have plus one times the quantity of U -3 and this is all equal to zero. And so now we see we have a common factor of U -3. So we can factor out that expo and we have the quantity of U minus three times the quantity of U plus one is equal to zero. And now recalling the zero factor property, we know we can set each individual expression equal to zero and solve for you. So starting with the first part of our expression where we have U minus three again, we can set this X expression equal to zero. And now solving for you, we need to add three to both sides and we get U is equal to positive three. And then repeating this with the other part of our expression, we have you plus one is equal to zero. Now subtracting one from both sides to solve for you, we have you is also equal to negative one. Now our next step is to substitute back in our original value for you so that we can start solving for X. So doing this, we can begin with our U is equal to three and we will now have X to the power of negative 1/5 is equal to three. And so now we need to solve for X. So we need to recall the power to a power rule for exponents which states 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 rule in mind, we know we can raise X to the power of negative five so that we will get rid of this exponents. And since we are raising this power, since we are raising the left side to the power of negative five, we need to do this to the right side as well. So we will have three raised to the power of negative five. And now just simplifying we have X is equal to three to the power of negative five, which simplifies to one divided by 243. And so now we can move on to repeating this with our other value that we found for you where we had U is equal to negative one. So now substituting back in X for you, we have X raised to the power of negative 1/5 is equal to negative one. And now just like before we need to raise both sides to the power of negative five. In doing this, we get X is equal to the quantity of negative one raised to the power of negative five, which just simplifies to negative one. So with this, we have found two values for X. So we can rewrite these solutions as a solution set. So we have the two values of X as negative one and one divided by 243. So with this, we have found two solutions for X and 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^-2/3+x^-1/3-6=0

Key Concepts

Exponents and Rational Exponents

Polynomial Equations

Factoring and Solving Techniques

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