Write in simplified radical form with at most one radical.
$\frac{\sqrt{x y^5}}{\sqrt[6]{x^2 y^3}}$
Assume that the variables represent positive real numbers.
Answer (1)
Rewrite the expression using fractional exponents.
Apply the power rule to the numerator and denominator.
Simplify the exponents.
Use the quotient rule to combine x and y terms.
Simplify the exponents further.
Rewrite the expression in radical form: y 2 6 x .
Explanation
Understanding the Problem We are asked to simplify the expression 6 x 2 y 3 x y 5 into a radical form with at most one radical. We assume that the variables x and y represent positive real numbers.
Rewriting with Fractional Exponents First, we rewrite the expression using fractional exponents. Recall that n a = a 1/ n . Thus, we have 6 x 2 y 3 x y 5 = ( x 2 y 3 ) 1/6 ( x y 5 ) 1/2 .
Applying the Power Rule Next, we apply the power rule to both the numerator and the denominator. The power rule states that ( ab ) n = a n b n . Applying this rule, we get ( x 2 y 3 ) 1/6 ( x y 5 ) 1/2 = x 2/6 y 3/6 x 1/2 y 5/2 .
Simplifying Exponents Now, we simplify the exponents. We have 2/6 = 1/3 and 3/6 = 1/2 . Thus, the expression becomes x 1/3 y 1/2 x 1/2 y 5/2 .
Applying the Quotient Rule We use the quotient rule to combine the x and y terms. The quotient rule states that a n a m = a m − n . Applying this rule, we get x 1/2 − 1/3 y 5/2 − 1/2 .
Simplifying Exponents We simplify the exponents. We have 1/2 − 1/3 = 3/6 − 2/6 = 1/6 and 5/2 − 1/2 = 4/2 = 2 . Thus, the expression becomes x 1/6 y 2 .
Rewriting in Radical Form Finally, we rewrite the expression in radical form. Recall that a 1/ n = n a . Thus, we have x 1/6 y 2 = y 2 6 x .
Final Answer Therefore, the simplified radical form of the given expression is y 2 6 x .
Examples
Imagine you are calculating the volume of a strange object that involves both square roots and sixth roots of some lengths. Simplifying such expressions, as we did here, allows you to combine these measurements into a single, manageable radical, making further calculations easier. This is useful in fields like engineering or physics where complex geometric calculations are common.
