Solve each equation. See Example 7. 3x^3/4 = x^1/2

Question

Accepted Answer

Hey, everyone here we are asked to solve for X in the following equation where we have four X raised to the power of 3/4 is equal to X raised to the power of one half. So to start solving for X here in this equation, we want to get rid of the radicals or in other words, get rid of the fraction exponents. So we need to recall the power to a power rule for exponents which tells us that to the power of N raised to the power of M is equivalent to A, to the power of N times M. Now with this rule in mind, we again want to get rid of the fraction exponents. So we can start by looking at the left side of our equation where we have the power of 3/4. Well, to get rid of the denominator, we need to raise the entire term to the power of the denominator which is four. And now since we raised the left side to the fourth power, we need to repeat this to the right side and raise the quantity of X raised to the one half power all raised to the power of four. So now we just need to simplify and so on the left side of our equation, we simplify this to 256 X cubed and this is equal to now X squared. So now we want to get all of our X terms on the same side of the equation. So we can go ahead and just subtract X squared from both sides and we get 256 X cubed minus X squared is equal to zero. And now we see that both terms here have a common factor of X squared. So we can factor out X squared and we get X squared times equality of 256 X minus one and this is still equal to zero. So now looking at the left side of our equation here, we need to find the X values that caused this side of the equation to be equal to zero. So that means we can set both parts of our expression individually equal to zero. So we have X squared is equal to zero and we also have 256 X minus one is equal to zero. So starting with X squared, well, just having X alone will also be equal to zero. So here we see that X is just going to be zero as one of our solutions. And now looking at our other equation where we have 256 X minus one is equal to zero. Here, we just need to solve for X. So first, we need to add one to both sides and we have 256 X is equal to one. And now we need to get rid of the coefficient of X. So we divide both sides by 256. And we see that another solution for X is one divided by 256. So now we have found two solutions for X which we can write as a total solution over here to the right. So we see that X is zero or one divided by 256. So with these two solutions, we are left with answer choice. A thanks for watching. I hope you found this video helpful and I'll see you again next time.

Solve each equation. See Example 7. 3x^3/4 = x^1/2

Key Concepts

Exponents and Radicals

Equating Exponents

Solving Polynomial Equations

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