Multiply: $\sqrt[6]{x} \cdot \sqrt[4]{y^3}$
Answer (2)
To multiply 6 x ⋅ 4 y 3 , we convert each radical to a rational exponent, find a common denominator, and combine the terms. The final result is x 12 2 y 12 9 .
;
x 12 2 y 12 9
Explanation
Understanding the Problem Let's break down this problem step by step. Our goal is to multiply two terms involving radicals by first expressing them with rational exponents that share a common denominator. This will allow us to combine them effectively.
Converting Radicals to Rational Exponents First, we need to express the radicals as rational exponents. Recall that n a = a n 1 . Therefore, we can rewrite the given expression as: 6 x = x 6 1 4 y 3 = y 4 3
Finding the Common Denominator Now, we want to find a common denominator for the exponents 6 1 and 4 3 . The least common denominator (LCD) of 6 and 4 is 12. So, we will convert both fractions to have a denominator of 12.
Rewriting Exponents with Common Denominator Let's convert the exponents to have the common denominator of 12: x 6 1 = x 6 × 2 1 × 2 = x 12 2 y 4 3 = y 4 × 3 3 × 3 = y 12 9
Multiplying the Expressions Now that we have the expressions with a common denominator, we can multiply them together: x 12 2 ⋅ y 12 9 This is the simplified form of the original expression using rational exponents with a common denominator.
Examples
Imagine you're calculating the area of a rectangle where the length is 6 x and the width is 4 y 3 . To simplify the area expression, you would follow the same steps as above, converting the radicals to rational exponents with a common denominator. This type of problem is also useful in physics when dealing with quantities that scale with fractional powers, such as in fluid dynamics or thermodynamics.
