Simplify the rational exponents

Question

[(64c^5/3)/(a^-1/5b^4/5)]^-3/2

Answer 1

[(64c^5/3) / (a^-1/5b^4/5)]^-3/2

 

First thing we need to do is to make the negative exponents into positive exponents.  We do this by flipping the terms associated with those negative exponents.

 

[(64c^5/3a^1/5) / (b^4/5)] ^-3/2 

 

We cube the terms in brackets.

 

[(b^4/5)³ / (64c^5/3a^1/5)³] ^1/2  =

 

[(b^12/5) / (64³c⁵a^3/5)] ^1/2 =

 

We 1/2 power the terms in brackets.  1/2 power is the same as square-root.

 

[(b^12/5)^1/2] / [(64³)^1/2(c⁵)^1/2(a^3/5)^1/2]]  =

 

Now we can break up the terms to simplify the expression.  Our goal is to cancel out squares and square-roots. 

 

[(b²)^1/2(b^2/5)^1/2] / [(64²)^1/2(64)^1/2(c²)^1/2(c³)^1/2(a²)^1/2(a^-7/2)^1/2]  =

 

[b*b^1/5] / [64*8*c*c^3/2*a*a^-7/4]  =

 

b^7/4 / (512a^-3/4c^5/2) =

 

[⁴√(b⁴) * ⁴√(b³) * ⁴√(a³) ] / [512 * √(c²) * √(c³)] = 

 

 

[b  ⁴√(b³)  ⁴√(a³)] / [ 512 c √(c³)]

 

 

I hope this helps.  If you need clearer solution, let me know.  I will scan my solutions and send them to you in a link so it will easier to read.