Solve each equation. (x-2)^2/3 = x^1/3

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Hello. Today we're going to be solving for X in the given equation. So what we are given is three X minus two raised to the power of 2/3 is equal to X to the power of 1/3. Now, before we begin this, let's just quickly recall the rule for rewriting square roots as exponents. Let's assume that we have the en route of X raised to the power of A. If we wanted to rewrite this as an exponent, we can rewrite this as X to the power of a over N. The power that's inside of the square root is going to be written in the denominator and the numerator of the exponent. And the number in front of the square root is going to be written as the denominator in the exponent. So with that being said in the current equation we can go ahead and rewrite the left hand side of the equation as the cube root of three x minus two squared. And on the right hand side we can rewrite X to the one third as the cube root of X. So now that we have this, we want to go ahead and remove the inside arguments of the cube roots in order to cancel out the cube roots, we're gonna raise both sides of the equation to the third power. This is going to cancel out the current cube roots and what we are left with is three X minus two squared is equal to X. Now we want to go ahead and simplify the left hand side of the equation. And in order to simplify the left hand side we're going to need to foil the three x minus two squared three x minus two squared is the same thing as saying three x minus two times three x minus two. And in order to foil this, we're going to multiply three X to three X, which is going to give us nine X squared, then multiply three X two negative two, which is going to give us negative six, X negative 2 to 3 X. Which is going to give us negative six X again and the negative 22 negative two, which is going to give us positive four, combining the like terms together. This is going to give us nine X squared minus 12 X plus four. And this is what the left hand side of the equation is going to simplify to, So what we currently have is nine X squared minus 12, X plus four is equal to X. And in order to solve for X, I want to go ahead and set this equation equal to zero. In order to do that, I'm going to attract X to both sides of the equation and this is going to give me nine X squared minus 13, X plus four is equal to zero. What I currently have is a fact arable, try no meal as the left hand side of this equation is going to factor 29, x minus four times x minus one equal to zero. Now, in order to solve for X, I'm gonna go ahead and take both factors of X. Set it equal to zero then solve and doing this is going to give me nine x minus four is equal to zero and x minus one is equal to zero. On the left hand side, I can add four to both sides of the equation and that's going to give me nine. X is equal to four, then I can divide both sides by nine and that's going to give me X is equal to 4/ on the right hand side. All I have to do is just add one to both sides of the equation and that's going to give me X is equal to one. So currently I have two potential solutions for X but I need to verify the solutions that we need to solve for. So what I can do is go ahead and take the solutions plug it into the original equation then solve. But I'm gonna be going ahead and plug it into this form of the equation. So if I want to verify X is equal to one, plugging this into the equation is going to give me the cube root of three times 1 -2 squared is equal To the Q Root of one. Now on the left hand side three times one is just going to give me three and three minus two is going to give me one. So what I have is the cube root of one squared is equal to the cube root of one. Now one squared on the left hand side is just going to give me one. So what I am left with is the cuba to one is equal to the cube root of one, which is a true statement. So X is equal to one is going to be a solution. Now we need to verify X is equal to 4/9, so plugging in X is equal to 4/9 is going to give me the cube root of three times 4/9 minus two squared is equal to three or the cube root of 4/9. Now on the left hand side we can raise three to be 3/ and we can cross simplify by reducing 3 to 1 and reducing 9 to 3. So what we are left with is 4/3 minus two and 4/ minus two is going to leave me with negative 2/3. So what I have is the q route of negative 2/ squared. And that is going to equal to the cube root of 4/9. Now, in order to simplify the inside of the cube root on the left hand side, I'm going to need to distribute the exponents to both the numerator and denominator and this is going to give me the cube root of positive 4/9. And that is going to equal to the cube root of 4/9. And this is a true statement, so X is equal to 4/9 is going to be a solution to the equation. So since both of our solutions work, X is going to equal to 4/9 and X is going to equal to one, and these are gonna be the solutions for X to the given equation. And with that being said, the answer to this problem is going to be D. So I hope this video helps you in understanding how to approach this problem and I'll go ahead and see you all in the next video.

Solve each equation. (x-2)^2/3 = x^1/3

Key Concepts

Exponents and Radicals

Equations and Solutions

Squaring and Square Roots

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