Problem

Solve each equation. Check your solutions. $$ 4 x…

Problem 62 Easy Difficulty

Solve each equation. Check your solutions.
$$
3 x^{23}-x^{13}-24=0
$$

Answer

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Precalculus

Algebra

Beginning and Intermediate Algebra

Chapter 11

Quadratic Equations, Inequalities, and Functions

Section 4

Equations Quadratic in Form

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Video Transcript

we have three x to the 2/3 power minus X to the 1/3. Minus 24 is equal to zero. So right, well, what makes us against kind of hard to solve, right? Only fractional exponents. If this was a quadratic, write something with the highest apologised to and I had entered your powers. Well, that could be pretty simple yourself. So how about if we let Variable Let's say we let you so we let you be equal two while X to the 1/3. Why extra? 1/3? Well, because notice how if we take extra 1/3 and we square that right extra 1/3 squared is where x to the 1/3 times two or exit a 2/3. So if we let you be extra 1/3 then this extra to 2/3 just becomes, um, three times you square because you is extra. 1/3 and 1/3 square is extra 2/3. Okay, we just have three. You school square. If we again let you be extra 1/3 and then just minus Well, actually, 1/3. That is you. We have three. You squared minus you minus 24. Okay, is equal to zero. So now solving this becomes world. I mean, a lot simpler right than this, because you look at this, we say, Look me factored right. So three, you squared minus U minus 24. If we go ahead and try and factor this what we get? Well, this isn't a factor. As let's see U minus three times, times three. You all right? That gives me the three you square times were all three. You are plus eight, right? Because three times eight is 24. So we get if we distribute here we get a u times three. You three, you square. Then you times eight is an eight. You and then a minus 90 you three times three. You is a minus 98 u minus nine. You that's minus one. You are the minus you. And then minus three times a positive rate is minus 24. This is factor. And then, um, was it's still gonna be have to be equal to zero. Okay, well, zero part property says right. If we have two things, we have a times B right that's equal to zero. Then either a or B must be equal is ill, right? So we have two factors being multiplied equals zero. This means that either U minus three is equal to zero. That means that you right is equal to three or we have wealth or three. U plus eight is equal to zero, while three u plus eight is equal to zero. That means that three U is equal Teoh Negative eight. So therefore you is equal to negative 8/3 or native 1/3. So you with equal to either three or you is equal to negative 8/3. Okay, so it's solved for you right in front of you is equal to three or native 1/3. But our original right, um, equation here. We're not stopping for you. We're solving for X, but what is But remember, we let you be equal toe extra 1/3 power so lovely south for you. Weaken, Let's go ahead and back something to and say, Well, you If you equal street then then while extra 1/3 right, Because you is extra. 1/3 with extra 1/3 is equal to three Okay off. Excellent. 1 30 to 3 because just, um que both sideways each side to the third power to get south for X. So you that acts is equal to well, three, um, 2 30 just three times, three times three, which is 27. So he So we have that X equal to 27. Okay. Yeah. What? Well, or, um, or we have that you is equal to negative 8/3 off us into native. 1/3. That means so. Or, um, again, you is extra 1/3. So we have X to the 1/3 right Is equal to Well, negative. A third's right in the same way to south for X. We just cube both sides. Take extra. 1/3 cube. We get back axe, you take native 1/3 cube, so X is equal to 27 or right. That's a sequel to while Negative 8/3 cube. Okay, so just cubing the top Cuban, the bottom. But we have while a negative toe, an ID power. This is gonna be negative. This is equal to negative. Um, a cube. So negative. 512 over 27. Okay, we got X is equal to 27 or X is equal to negative. 512 over 27. Okay. Well, then we have to go and check. Both of these solutions. Made a substitution here. We could have picked up a extraneous social. So we have to go and check that these two solutions we found are actually ballot solutions for X in our original equation. Okay, so first I'm gonna go ahead and check. X is equal. 27 x is equal 27. Or then we have. Then we have three x 2 to 2/3. We have three times 27 to the 2/3 minus X to the 1/3. So minus 27 to the one third, um, and then minus 24. Right. That's our original equation right here. And that again should be equal to zero. This should be equal to zero. Okay, well, is it? Well, we have three times. 27 to the 2/3. That's, um Well, that's three times three cube. Right to the 2/3. You could write this as three times 27 is just three Q Q three, Q 2 to 2/3. So that a power to power, We just get back 27. So three times 27 to the 2/3 if we just get back 27 then just minus of 27 for the ones that the cube root off 27 which is three there's 27 minus three and then minus 24 should be equal to zero, which it is. Right, cause 27 minus three is 24 24 miles, 24 0 So you that zero is equal to zero, which is true. So therefore, our value of X equals 27 is a god of solution. So 27 is a solution. How about, um X is equal to negative. 512 over 27. Saying way to check this. We have three again times extra, 2/3 or three times negative. 512 over 27 to the 2/3. Okay, then minus extra. 1/3. So minus muscle minus and negative. Careful with the mind to negative. Minus negative. 512 over 27 to the 1/3. Minus 24 should be equal to zero. Well, is it? Well, we told this here, This becomes three times. Well, negative 5/27 is native 1/3. So this is really negative. Eight third's, um, cubed. Right? But then that is then to the 2/3 power. Okay. And then minus again, This is going to be thought of as minus, um, negative. 5 12/27 That is negative. Eight third's cube. Okay. And then we're raising that to the 1/3 power and then minus 24 should be equal to zero. Okay, when we get here well, negative 8/3 cube to the 2/3 again. That's three times 2/3 which is just, um, just, uh, to write. So we get three times. We could think of this as just, um Well, negative 8/3. Uh, squared. OK, which is gonna be 64 over threes we get. This is equal to 64. Um, we're over three because or nine the three cancels out, just get back 64/3 and then get while plus might minus and negative. So plus, while native, 8/3 right, this is just, um, to the right cube to the 1/3 that just to the first power, we just get back at plus 8/3 and then minus 24 should be equal to zero which it is, right. 64 3rd plus 8/3. Um, well, that's actually equal to 24. Minus 24 is equal to zero. So this week we get back. Zero is equal to zero. So the solution checks out as well. So hence our solution set is the set containing these two values, namely negative. 512 over 27 27. All right, All right, take it.

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