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^2 - x - 4)^(3/4) - 2 = 6

Accepted Answer

Welcome back. I'm so glad you're here. We're given this equation, we have four X squared minus X plus four. All raised to the 3/5 plus eight equals zero. And we're to solve for X. Well first let's bring that eight over to the right hand side of the equation so that we have the radical isolated over here. We've got four X squared minus 24 X plus four raised to the 3/ equals negative eight on the right hand side. And now to get rid of that radical we want to take the whole thing and raise it to the five thirds and what we do to one side we're going to do to the other. So when we've got an exponent raised to an exponent while those multiply. So 3/5 times five thirds. That's one. So now the left hand side is four X squared minus 24 X plus four. And the right hand side. Well that's not as beautiful. It's negative eight raised to the five thirds. So let's go off to the side for a moment and let's do some work on negative eight to the five thirds. To simplify that. Negative eight to the five thirds is the same thing as the third rate of negative eight to the fifth and negative eight to the fifth. Well that is negative 32,768 obviously and we take the third root of that and that is going to be a negative 32. Well that is a much nicer number to have on that right hand side of the equation. So we've got four X squared minus 24 X plus four equals negative 32. Let's bring that back over to the left hand side so that we can set this thing equal to zero and perhaps do some factoring. So now we've got four X squared minus 24 X plus 36 4 plus 32 is 36 equals zero. And now everything there is divisible by four. So let's divide everything by +44, X squared divided by four is X squared negative 24 X divided by four is negative six. X 36 divided by four is plus nine and zero divided by four is zero. Great. Now let's do some factoring So X. What gives us X squared X times X. What gives us what two factors give us a positive nine When multiplied in a negative six When added together, that will be a minus three minus three. And now we've got factors, we can set those equal to zero. Well they're the same thing. So we can say X -3 equals zero. We can add three double sides and we get an answer of X equals three. Woo hoo. We've got an answer X equals three. That's right at the bottom. Let me write it over here to the side, X equals three. So it's not right at the bottom. And now we need to check our solution. So let's go back in with the original equation. We've got four X squared minus 24 X plus four all raised to the 3/5 plus eight equals zero. And now we'll plug in a three for all our exes. So it will be four times three squared minus times three plus four, raised to the 3/5 plus eight equals zero and three squared is nine. So we've got four times nine minus times three is 72. So minus 72 plus four raised to the 3/ plus eight equals +04 times nine is 36. So 36 minus 72 plus four. When we do all of that addition and subtraction. That will give us a negative 32 raised to the 3/5 plus eight equals zero. And hey that negative 32 is familiar. Well negative 32 to the 3/5 is the same as the fifth root of negative 32 to the third plus eight equals zero. And when we take negative 32 to the third that is the same as negative 32,768. Well that's familiar. Plus eight equals zero. Now when we take the fifth root of negative 32,768 that is negative eight. So we've got negative eight plus eight equals zero and yes that is absolutely true. So our solution of X equals three. That works. We look at our answer choices and that matches with answer choice B. Well done, we'll catch you on the next one.

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