Solve each equation with rational exponents in Exercises 31–40. C...

Question

Solve each equation with rational exponents in Exercises 31–40. Check all proposed solutions.

(x - 4)^(3/2) = 27

Accepted Answer

Welcome back. I'm so glad you're here. We have this radical equation. We've got three X plus four all raised to the 3/4 and that equals eight. Okay well now we want to get rid of that to the 3/4 which is a radical. And so we're going to take that left hand side of the equation and raise it to the four thirds. But what we do to one side we have to do to the other. So we're going to raise eight to the four thirds as well. So now on the left when we have an exponent raised to another exponent we multiply those exponents. So 3/4 times four thirds. Well that's just one. So the left hand side of the equation is three X plus four. And the right hand side is well let's just eight to the four thirds. So let's go off to the side here and do a little bit of work to simplify Eight to the four thirds. So eight to the four thirds is the same as the cubed root of eight to the fourth and the cubed root of eight to the fourth is the same as the cubed root of Well eight to the fourth is 4096. And when we take the cubed root of 4096 that is 16. Well that is a much nicer number. So we now have three X plus four equals 16. Okay let's subtract four from both sides to get that three X by itself. So we have three X equals 16 minus four is 12. And let's divide both sides by three to get X by itself. And that gives us X equals four. Well great. We've solved for X. And now we are asked to perform the checking of the solutions to make sure we did it correctly. So let's substitute an X. Or a four back in for X. So in three X plus four raised to the 3/4 equals eight. It will now be three times four plus four to the 3/4. It's a lot of threes and fours equals eight. So three times four is 12. So 12 plus four to the 3/4 equals eight and 12 plus four is 16. So 16 to the 3/4 equals 8. 16. That's familiar. 16 to the 3/4 is the same thing as the 4th root of 16. Cute equals eight And 16. Cute. Well that is 4,096. That's familiar. So the fourth root of 4096. Well that is eight. So we get an answer of eight equals eight. That checks out. So X does in fact equal four. We look at our answer choices and that matches answer choice. D Well done. We'll catch on the next one

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