13. For what x0 does x^(x^x) = (x^x)^x? Give reasons for y...

Question

13. For what x>0 does x^(x^x) = (x^x)^x? Give reasons for your answer.

Accepted Answer

Welcome back, everyone. Find all positive real numbers X satisfying X to the power of X cubed equals X cubed to the power of X. So let's begin analyzing this equation X to the power of X cubed is equal to x cubed to the power of X. What we can do is simply rewrite the right-hand side using the properties of exponents. So that's X raises the power of 3 multiplied by X or basically 3 X. Now, what we can do is simply divide both sides by. X to the power of 3 X, right? So we're going to get X to the power of X cubed. According to the properties of exponents, we're just subtracting the exponent 3 X, right? And this is equal to 1. So whenever we divide two terms with the same base with different exponents, we are subtracting those exponents. And now we have an equation. Knowing that X is greater than 0, so we have two possible cases. Either our exponent is equal to 0 because whenever we raise a number to an exponent of 0, we get 1, meaning the first case is that X cubed minus 3 X is equal to 0, or X is simply 1 because whenever we raise 1 to any exponent, we're going to get 1. So either X cubed minus 3 X is equal to 0 or X is equal to 1. So we already have one of the solutions. X is greater than 0. X is equal to 1 is a possible solution, and now we want to solve X cubed minus 3X equals 0. And we can perform factorization X multiplied by X2 minus 3 is equal to 0. Now, one of the possibilities is to have X of 0. Using the Zero product property. And it doesn't satisfy the condition that X must be greater than 0, so we can exclude it. And now for the other factor, we can factor it using the difference of squares, right? So, we're going to have square roots. Of X2 we're going to write x minus square root of 3 multiplied by. X Plus square root of 3, right? And this is going to be equal to 0. So from here we have two possible solutions. X is either positive or negative square root of 3. And because the X is greater than 0, we're going to take the positive solution only. So we have two possible solutions. X is equal to 1, or X is equal to square root of 3. Those would be the final answers for this problem. Thank you for watching.

13. For what x>0 does x^(x^x) = (x^x)^x? Give reasons for your answer.

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