Question 1204250
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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: {{{matrix(1,3, 125^x, "=", a*sqrt(b))}}}.
                 A different "spin" on this is: {{{matrix(10,3, 625^x, "=", 64, (125 * 5)^x, "=", 64,
125^x * 5^x, "=", 64, (5^3)^x * 5^x, "=", 64, 5^(3x) * 5^x, "=", 64, 5^(3x + x), "=", 64, 5^(4x), "=", 64, (5^(3x))^(4/3), "=", 64, ((5^3)^x)^(4/3), "=", 64, (125^x)^(4/3), "=", 2^6)}}} 
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 = 4<sup>3</sup>. So, as you may know and can clearly see, 64 can either be expressed as 8<sup>2</sup>, 2<sup>6</sup>, or 4<sup>3</sup>.
                                                  {{{matrix(1,3, (125^x)^(4/3), "=", (2^(9/2))^(4/3))}}}
                                                       {{{matrix(1,3, 125^x, "=", matrix(2,1, " ", 2^(9/2)))}}} ----- EQUATING BASES, since EXPONENTS are the same
                                                       {{{matrix(2,3, 125^x, "=", matrix(2,1, " ", 2^(4 + 1/2)), 125^x, "=", (2^4)(2^(1/2)))}}}
                                                       {{{matrix(1,3, 125^x, "=", 16sqrt(2))}}} -------- {{{matrix(2,3, 2^4, "=", 16, matrix(2,1, " ", 2^(1/2)), "=", sqrt(2))}}}
As "a" and "b" are being sought from the equation: 125<sup>x</sup> = a√b, and {{{matrix(1,3, 125^x, "=", 16sqrt(2))}}}, then IN THIS CASE, <font color = red><font size = 4><b>a = 16</font></font></b>, while <font color = red><font size = 4><b>b = 2</font></font></b>

BTW, {{{16sqrt(2)}}} is the SAME as {{{8sqrt(8)}}}. Check this yourself!! And, if base 4 (4<sup>3</sup>) 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."</pre>