In Exercises 1–38, solve each radical equation.

(3x - 6)¹/³ + 5 ...

Question

In Exercises 1–38, solve each radical equation.

(3x - 6)¹/³ + 5 = 8

Accepted Answer

Welcome back. I am so glad you're here. We are told in the given radical equation to solve for X. Our radical equation is open parentheses five, X minus 11 closed parentheses raised to the one third power plus three equals seven. Our answer choices are answer choice. A, the solution set negative 15, answer choice B, the solution set 15, answer choice C the solution set 75 answer choice D the solution set negative 15 15. All right. When we are solving for X, we want to get X totally by itself on one side of the equal sign and essentially move over all of the other numbers and operations that are on the same side as the X. And we've got a lot here inside of parentheses. X is being multiplied by this five. It's having 11, subtracted from it. And then it's being raised to the one third power and then it has the plus three on the outside of the parentheses. And all of that needs to be scooted over to the other side. But how do we do that? Well, we recall from previous lessons that we need to follow the reverse order. Of the order of operations. We recall that acronym for parentheses, exponents, multiplication and division, addition and subtraction. Normally we work left to right now, we're going to work right to left to get X by itself. So we note that X is inside of parentheses. So we want to move over anything that's outside of parentheses first and we start on the right. So we start with addition and subtraction and we see that outside of the parentheses, there's this three being added. So to move the three to the other side of the equal sign, we're going to do the reverse of the operation that's being done to it, it's being added. So we're going to subtract three and what we do to one side of the equation we must do to the other. So we're subtracting three from both sides of the equation on the left, the positive three minus three, that's zero. So all we have is our open parentheses, five X minus 11, closed parentheses raised to the one third power and that's equal to on the right and we have seven minus three, which is a positive four. So we've done all of the addition and subtraction outside of the parentheses. There's no multiplication or division outside of the parentheses. Next, we see an exponent that what's in the parentheses is being raised to. And this exponent is interesting because this is a fraction, this is to the one third power. And we recall from previous lessons that when we have something being raised to a fraction. In this case, the one third power, we are going to take the denominator of that fraction exponent. And that's going to be the rut that is taken of our number. So we're taking the third rut of what's inside of the parentheses. That's where the radical part of this radical equation comes from. Now, in order to undo taking the third root of something, we're going to raise it to the third power and what we do to one side, we do to the other. So we're taking both sides of this equation and ra raising it to the third power. Noting that when we have a power raised to a power, we would be multiplying them. So taking one third multiplied by three, that's one. And that's saying that our quantity of five X minus 11 is raised to the first power, which is just five X minus 11. And that's how we got rid of that power on the left. Now, on the right, we have four raised to the third power and four cubed is 64. Well, now by doing the exponents, we've gotten rid of the parentheses, but there are still things left to do. So we start back over on the right of our order of operations with addition and subtraction. We look for addition and subtraction and here we have subtraction this minus 11. So we're going to do the opposite operation of subtraction. We're going to add 11 and what we do to one side, we do to the other on the left negative 11 plus will be zero. So all we have is a positive five X that's equal to on the right hand side, 64 plus 11 is a positive 75. All right, we have finished the addition and subtraction step again. Now we look for multiplication and division and we do have multiplication. We have five multiplied by X. So we do the opposite of multiplication, which is division. We're going to divide both sides by five. And on the left five divided by five is one, we have one X which is just X and that gives us our X by itself, which is what we wanted. Yay, that's equal to on the right hand side 75 divided by five, which is a positive 15. And we have completed our solving for X X is equal to 15. We look at our answer choices and that matches with the answer choice. B well done. We'll catch you on the next one.

In Exercises 1–38, solve each radical equation. (3x - 6)¹/³ + 5 = 8

Key Concepts

Radical Equations

Isolating the Variable

Extraneous Solutions

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