Final answer:
To simplify the expression (125x^4y^3/4)^1/3 ÷ (xy)^1/2, you cube root the numerator, take the square root of the denominator, and then divide the two, resulting in the expression 5x^(5/6)/y^(1/4).
Step-by-step explanation:
Let's simplify the expression (125x^4y^3/4)^1/3 ÷ (xy)^1/2 step by step.
First, we apply the exponent of 1/3 to the fraction inside the parentheses:
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- (125)^1/3 = 5 because 5^3 = 125
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- (x^4)^1/3 = x^(4/3) because we multiply the exponents
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- (y^3/4)^1/3 = y^(1/4)
So, the expression inside the parentheses simplifies to 5x^(4/3)y^(1/4).
Next, we simplify the expression (xy)^1/2:
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- x^1/2 and y^1/2 are the square roots of x and y, respectively
Now the entire expression becomes (5x^(4/3)y^(1/4)) ÷ (x^1/2y^1/2).
To divide, we subtract the exponents of like bases:
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- x^(4/3 - 1/2) = x^(8/6 - 3/6) = x^(5/6)
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- y^(1/4 - 1/2) = y^(1/4 - 2/4) = y^(-1/4)
Since y^(-1/4) is in the denominator due to the negative exponent, our final answer is 5x^(5/6)/y^(1/4).