Solve each equation. See Example 7. x^2/3 = 2x^1/3

Question

Accepted Answer

Hey, everyone here we are asked to solve for X in the following equation, X to the power of two thirds is equal to five X to the power of one third. Now, before we begin solving for X, 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 now with this rule in mind, we can look back at our given equation and we want to begin by eliminating the radical in both of our terms here. So that means we need to get rid of the fraction in our exponents. So starting by looking at the left side of our equation, we have the power of two thirds. Well, to get rid of the denominator, we need to multiply by the denominator, which means we need to raise this term to the power of the denominator which is three. So we have the quantity of X raised to the two thirds, power all raised to the power of three. And now since we did this to the left side of our equation, we also need to do it to the right side of our equation. So now simplifying we have just X squared on the left side is equal to 125 X to the power of one, which is just X, the same as just writing X. So we have X squared is equal to 125 X. And here we see, we have gotten rid of the fractions in the exponents. So now we just want to get all of our X terms on the same side of the equation. So we can just move over this 125 X from the right side to the left side by subtracting 125 X from both sides of our equation. So now rewriting, we have X squared minus 125 X is equal to zero. So now we see that both terms on the left side have an X in them. So we can factor out this common factor. So we have X times the quantity of X minus 125 and this is still equal to zero. And now we just need to find what values of X set this expression equal to zero. And that means we can apply the zero factor property which allows us to set each part of the expression or each factor equal to zero individually. So we can start with the outside term, which is just X and set it equal to zero. Well, this immediately gives us a solution of zero. So we can move on to the inside expression we have, which is X minus 125. And again, we can set this equal to zero. And solving for X, We need to add 125 to both sides. And we see that a second solution of X is 125. So we have found two values or two solutions for X which are just zero and positive 125. So with this, 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 Example 7. x^2/3 = 2x^1/3

Key Concepts

Exponents and Radicals

Isolating Variables

Quadratic Equations

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