[(x^3/2)^4/3]*(x^1/4)/x

Let us simplify:

First we will simplify the first term:

[(x^3/2)^4/3]

We know that: (x^a)^b = x^(a*b)

==> x^3/2)^4/3= x^(3/2 * 4/3) = x^4/2 = x^2

==> (x^2)*x^1/4 / x

Now we know that: x^a * x^b = x^(a+b)

=> x^2 * x^1/4 = x^(2+1/4) = x^9/4 =

==> (x^9/4)/x

==> Now we know that: x^a/x^b = x^(a-b)

==>( x^9/4)/ x = x^(9/4 - 1) = x^5/4

**Then the final form is: x^5/4**

To write[ (x^3/2)^(4/3)]*(x^1/4)/x as x x raise to the power.

{(x^3/2)^(4/3)} (x^(1/4) /x

=(x ^((3/2)*(4/3)) * x^ (1/4) * x^(-1)

=(x^4/2)*x^(1/4) *(x^-1) , as (x^m)^n = x^(mn)

=(x ^2)(x^1/4)(x^-1)

=x^(2+1/4-1) , as (x^l)(x^m)(x^n) = x^(l+m+n).

=x^(1+1/4)

=x^(5/4)

Therefore [( x^3/2)^4/3]*(x^1/4)/x = x^(5/4)

We'll write the first term of the product from numerator as:

(x^m)^n = x^(m*n), where m = 3/2 and n = 4/3

(x^3/2)^4/3 = x^(3*4/2*3)

We'll simplify and we'll get:

(x^3/2)^4/3 = x^2

Now, we'll solve the product from numerator:

[(x^3/2)^4/3]*(x^1/4) = (x^2)*(x^1/4)

Since the bases are matching, we'll add the exponents:

(x^2)*(x^1/4) = x^(2 + 1/4)

x^(2 + 1/4) = x^9/4

Now, we'll solve the ratio:

x^9/4/x = x^(9/4 - 1)

**x^(9/4 - 1) = x^5/4**