Simplify using the exponent law. Express your answer using positive exponents only.

[4x^1/3]^1/2 [9x]^-3/2 / [27x]^-1/3

= 2(x^(1/6) ((1/27) x^(-3/2) (3) (x^(1/3)

= (6/27) x^(1/6-3/2+1/3)
= (2/9) x^-1
= 2/(9x)

To simplify the expression using the exponent law, we need to apply the rules for exponents. Let's break down the expression step by step:

First, let's simplify the expression inside the square brackets: [4x^(1/3)]^(1/2).

According to the exponent law, when we raise a power to another power, we multiply the exponents. Hence, we multiply (1/3) and (1/2) to simplify:

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

Next, let's simplify 4^(1/2):

4^(1/2) is the square root of 4, which is 2. Therefore,

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

Now, let's simplify (x^(1/3))^(1/2):

According to the exponent law, when we multiply two exponents with the same base, we add the exponents. Therefore:

(x^(1/3))^(1/2) = x^((1/3) * (1/2)) = x^(1/6)

So, the expression inside the square brackets becomes:

[4x^(1/3)]^(1/2) = 2x^(1/6)

Now, let's simplify the expression inside the parentheses: [9x]^(-3/2).

According to the exponent law, when we have a negative exponent, we can move the term to the denominator and change the sign of the exponent. Therefore:

[9x]^(-3/2) = 1/[9x]^(3/2)

Lastly, let's simplify the expression in the denominator: [27x]^(-1/3).

Just like before, we move the term to the denominator and change the sign of the exponent:

[27x]^(-1/3) = 1/[27x]^(1/3)

Now, we have the following expression:

(2x^(1/6) * 1/[9x]^(3/2)) / (1/[27x]^(1/3))

To divide fractions, we can multiply the first fraction by the reciprocal of the second fraction. Therefore:

(2x^(1/6) * 1/[9x]^(3/2)) * ([27x]^(1/3)/1)

Now, let's apply the exponent law to simplify each term:

1/[9x]^(3/2) = 1/(9^(3/2) * (x^(3/2)) = 1/(9^(3/2) * x^(3/2))

[27x]^(1/3) = (27^(1/3) * (x^(1/3)) = 3x^(1/3)

Substituting these simplified expressions back into the main expression, we get:

(2x^(1/6) * 1/(9^(3/2) * x^(3/2))) * (3x^(1/3)/1)

To multiply with the same base, we add the exponents:

2 * x^(1/6 - 3/2 + 1/3)

Now, let's simplify the exponent:

1/6 - 3/2 + 1/3 = (1 - 9 + 2)/6 = -6/6 = -1

Therefore, the simplified expression is:

2x^(-1)

Expressing the answer using positive exponents, we move the term to the denominator and change the sign of the exponent:

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

Hence, the simplified expression with positive exponents is:

2/x