In Exercises 1–38, solve each radical equation.

(2x + 3)¹/⁴ + 7 ...

Question

In Exercises 1–38, solve each radical equation.

(2x + 3)¹/⁴ + 7 = 10

Accepted Answer

Welcome back. I am so glad you're here. We're told in the given radical equation to solve for X. Our radical equation is open parentheses four X minus three, closed parentheses raised to the 1/4 power plus two equals seven. Our answer choices are answer choice. A the solution set negative 157. Answer choice B the solution set answer choice C the solution set negative and answer choice D the solution set 628. All right. When we are solving for X, we want to get X by itself on one side of the equal sign. And that means we have to get everything else over to the other side. We've got the four being multiplied, the three being subtracted this 1/4 power it's being raised to. And then this plus two, we have to move all of those over to the right hand side. And in order to do that, we need to follow essentially the reverse of the order of operations. So we recall from previous lessons, the order of operations through the acronym peda parentheses, exponents, multiplication and division, addition and subtraction. And we work our way from left to right. But when we're getting X by itself, we're working our way from right to left. And it's important to note here that the X is inside of parentheses. Parentheses would be last in our reverse order of operations that we're following. So we're going to start by looking for anything outside of the parentheses. And the only operation happening outside of the parentheses is the plus two. So in order to get that positive two to the other side of the equal sign, we're going to do the reverse of the operation happening to it. So the two is being added, we're going to subtract two and what we do to one side we must do to the other side. So we're subtracting two from both sides of the equation on the left that will leave us with just our open parentheses. Four X minus three, closed parentheses raised to the 1/4 because two minus two is zero and then that will be equal to on the right hand side, seven minus two is a positive five. And so now we have done all of the addition and subtraction outside of the parentheses, the multiplication and division outside of the parentheses. There isn't any. But now we deal with an exponent outside of the parentheses. This exponent is interesting because this exponent is a fraction. This is the radical part of our equation. We recall from previous lessons that taking anything, I'll just call it X raised two, the 1/4 power is the same as taking the fourth root of X. And so, in this case, in order to do the opposite of this raising to the 1/ power, in order to get that raised to the 1/4 power to the other side, what we're going to do because we're really taking the fourth rut of X, we are going to raise the whole thing on both sides to the fourth power. That's how you would undo the fourth threat. You take it to the fourth power. And noting here that taking 1/4 the power raised to the power of four, we would be multiplying those and 1/4 multiplied by four is one. So the left hand side would just be four, X minus three, all raised to the first power, which is just four X minus three. And that equals on the right hand side, five race to the fourth power which is 625. All right. So we've gotten the exponents taken care of, but now we're not inside of parentheses anymore. So we start back over on the right hand side of the order of operations. We look for any addition or subtraction again and we have subtraction, we're subtracting that three. So we do the opposite of subtraction, which will be addition, we add three and what we do to one side, we do to the other on the left, we'll just have four X because a negative three plus three is zero. And then on the right, 625 plus three is 628. So we have four, X equals 628. We almost have X by itself. There's no more addition or subtraction. Now we look for multiplication or division, we have multiplication, we have the four multiplied by the X. And so the opposite of multiplication is going to be division. We're going to divide both sides by four. And on the left, the four divided by the four, that's just one. So we have one X which is just X, which is what we want. X is by itself. Yay. And on the right 628 divided by four is 157. So we have completed our reverse of the order of operations because X is by itself X equals 157. And that matches with answer choice B well done. We'll catch you on the next one.

In Exercises 1–38, solve each radical equation. (2x + 3)¹/⁴ + 7 = 10

Key Concepts

Radical Equations

Isolating the Variable

Extraneous Solutions

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