(x+y)^2/3(x-y)^2/3÷√x+y×√(x-y)^3 6

Jasonbradley

3 min read

·

Jun 17, 2024

34

Share

Simplifying the Expression: (x+y)^(2/3)(x-y)^(2/3) ÷ √(x+y) × √(x-y)^3

This problem involves simplifying a complex expression with fractional exponents and radicals. Let's break it down step by step:

Understanding the Components

• Fractional Exponents: A fractional exponent like (2/3) indicates a root and a power. In this case, (x+y)^(2/3) means the cube root of (x+y) squared.
• Radicals: The symbol √ represents the square root.
• Division and Multiplication: Remember the order of operations (PEMDAS/BODMAS). Division and multiplication are performed from left to right.

Simplifying the Expression

1. Combine like terms: Notice that (x+y)^(2/3) and (x-y)^(2/3) can be grouped together.

   (x+y)^(2/3)(x-y)^(2/3) ÷ √(x+y) × √(x-y)^3 = [(x+y)^(2/3)(x-y)^(2/3)] ÷ √(x+y) × √(x-y)^3

2. Simplify the numerator: Use the rule that states (a^m)(a^n) = a^(m+n).

   [(x+y)^(2/3)(x-y)^(2/3)] ÷ √(x+y) × √(x-y)^3 = [(x+y)^(2/3 + 2/3)(x-y)^(2/3 + 2/3)] ÷ √(x+y) × √(x-y)^3 = [(x+y)^4/3 (x-y)^4/3] ÷ √(x+y) × √(x-y)^3

3. Rewrite the radicals in exponential form: √a = a^(1/2)

   [(x+y)^4/3 (x-y)^4/3] ÷ (x+y)^(1/2) × (x-y)^(3/2)

4. Apply the division rule for exponents: a^m ÷ a^n = a^(m-n)

   [(x+y)^(4/3 - 1/2) (x-y)^(4/3 - 3/2)] = (x+y)^(5/6) (x-y)^(-1/6)

5. Express negative exponents in the denominator: a^(-n) = 1/a^n

   (x+y)^(5/6) / (x-y)^(1/6)

Final Simplified Expression

The simplified form of the given expression is (x+y)^(5/6) / (x-y)^(1/6).