express as a single fraction:X to power -1/3 times Y to power 1/3 minus X to power 2/3 times Y to power -2/3.
Q4(5X)power a+4(5X)power 1 soltn; add the powers since we are multipling same bases,indice=(5X)power a+4+1=(5X)power a+5
x^(-1/3) * y^(1/3) - x^(2/3) * y^(-2/3)
= (y^1/3 / x^1/3) * (x^2/3 / y^2/3)
= (x^2/3 y^1/3) / (x^1/3 y^2/3)
= x^1/3 / y^1/3
= (x/y)^(1/3)
To express the given expression as a single fraction, we need to find a common denominator for the exponents and combine the terms.
Let's start by rewriting the expression with a common denominator.
X^(-1/3) * Y^(1/3) - X^(2/3) * Y^(-2/3)
To find a common denominator for the exponents, we need to find their least common multiple (LCM) of 3.
The LCM of 3 and 3 is 3. Therefore, we'll rewrite the exponents in terms of the common denominator.
X^(-1/3) = X^(-1/3) * (Y^2/2) / (Y^2/2) = (Y^2)^(1/3) / (X^(1/3) * Y^(2/3))
X^(2/3) = (X^2)^(1/3) = (Y^2/2)^(1/3) / (Y^(2/3))
Now, let's combine the terms with the common denominator.
(X^(-1/3) * Y^(1/3)) - (X^(2/3) * Y^(-2/3)) = (Y^2)^(1/3) / (X^(1/3) * Y^(2/3)) - (Y^2/2)^(1/3) / (Y^(2/3))
Now, we'll combine the numerators. Since the base is the same (Y^2), we can subtract the exponents.
(Y^2)^(1/3) - (Y^2/2)^(1/3) = Y^(2/3) - (Y^2)^(1/3) / (Y^(2/3))
Finally, the expression in the form of a single fraction is:
(Y^(2/3) - (Y^2)^(1/3)) / (X^(1/3) * Y^(2/3))
Note: It's important to simplify any further if possible.