Let's follow the steps to simplify the expression (x^(-3)*y^((1)/(3)))^((3)/(2))
Step 1: Rewrite the term
The first step would be to convert the negative exponent in x^(-3) to positive. In the language of exponents, negative powers indicate a reciprocal. Hence, x^(-3) becomes 1/x^3. Thus the original expression can be rewritten without using a negative exponent as follows:
(x^(-3) * y^(1/3))^(3/2) = (1/x^3 * y^(1/3))^(3/2)
Step 2: Apply the power rule of exponents
Power rule of exponents states that when raising a power to a power, we multiply the exponents so we multiply exponents inside and outside the parentheses.
(1/x^3 * y^(1/3))^(3/2) = 1/x^((3)(3/2)) * y^((1/3)(3/2)) = 1/x^(9/2) * y^(1/2)
Step 3: Simplify the expression
The fraction 9/2 in the exponent of x is not a simplified form, as 9/2 = 4.5. Writing the exponent as an improper fraction simplifies it in terms of square root of x.
Next, write y^(1/2) as √y. Now, it should look like:
1/x^(4.5) * √y
Step 4: Write in a convenient form
For visual simplicity and understanding, we write x^(4.5) as √(x^9) and then, we reciprocate it to get rid of the term from the denominator:
1/x^(4.5) * √y = √y / √(x^9)
Finally, apply the power rule for the second time outside of the parentheses. This will be last step to simplify the expression:
( √y / √(x^9) ) ^3 = (√y)^3 / (√(x^9))^3
The final, simplified form of the original expression (x^(-3)*y^((1)/(3)))^((3)/(2)) is:
(y / √x)^3
This final expression is the simplified form of the original with all variables as positive real numbers, and without using negative exponent.