SOLUTION: Let x be a real number such that 625^x=64 Then 125x=a√b. What are a and b?

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Question 1204250: Let x be a real number such that 625^x=64 Then
125x=a√b. What are a and b?
Found 3 solutions by MathLover1, ikleyn, MathTherapy:
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!



Let be a real number such that
Then What are and ?

We have the equation
To simplify the equation, we can express both sides with the same base. We know that can be expressed as and can be expressed as .
Rewriting the equation, we have

Using the property of exponents, we can simplify further:

To find , we need to equate the exponents:


Now, solving for:


Now, we can calculate the value of using the value of we found:

since , we have
then

since , then

so, comparing to , we have




Answer by ikleyn(52855)   (Show Source): You can put this solution on YOUR website!
.
,  the problem in the post is written incorrectly.
The correct formulation is as follows 

    Let x be a real number such that  = 64.

    Then   = .  Find "a" and "b".


,  the  " solution "  by  @MathLover1 is  TOTALLY  WRONG.
I came to bring a correct solution. 

              


We are given   = 64.

It is the same as to write

     = 64.    (1)


Then

     =  =  = now replace    by  64,  based on  (1)  = 64^(3/4) = 8^(3/2) = .


Thus the problem is just solved.  a = 8;  b = 8.     ANSWER



              


If 625^x = 64,  then  x*log(625) = log(64), x =  = 0.646014837  (approximately).


Next,   =  = 22.627417 (approximately);   

        = 22.627417  (the same value).   Check is completed and confirms the answer.

Solved.


///////////////////////


For safety of your mind, simply ignore the post by @MathLover1.

Observing her activity at this forum during years, I learned that she DOES NOT know Math, at all
(except of very local pieces).

Her method to work at this forum is to re-write from other tutors or from web-sites
(if she find an appropriate source) - without any reference, naturally.

If she does not find the source to re-write from, she writes any gibberish,
without hesitation.


What others think about her creations, she doesn't care.


Such "tutors" should not be allowed approaching to Math education closer than a cannon shot.



Answer by MathTherapy(10555)   (Show Source): You can put this solution on YOUR website!
Let x be a real number such that 625^x=64 Then
125x=a√b. What are a and b?

That woman is TOTALLY LOST and CLUELESS!!! Who ever heard of EQUATING the EXPONENTS of 2 exponential expressions when they have DIFFERENT bases? 

It appears, as TUTOR @IKLEYN states, that the correct equation is: .
                 A different "spin" on this is:  
As seen directly above, 2 is being used as the BASE of 64 i/o 8, as Tutor @IKLEYN did. But, 4 could've also been used
since 64 = 43. So, as you may know and can clearly see, 64 can either be expressed as 82, 26, or 43.
                                                  
                                                        ----- EQUATING BASES, since EXPONENTS are the same
                                                       
                                                        -------- 
As "a" and "b" are being sought from the equation: 125x = a√b, and , then IN THIS CASE, a = 16, while b = 2

BTW,  is the SAME as . Check this yourself!! And, if base 4 (43) is used for 64, another set of values for "a" and "b" 
- maybe a different set - will ensue! You may want to try that one on your own since 2 of us already used bases 8 and 2. Okay?

Therefore, there is no UNIQUE INTEGER set of values for "a" and "b."
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