Final answer:
To simplify ³√x³ / ⁵√x², rewrite the radicals as fractional exponents. The cube root of x³ simplifies to x, and the fifth root of x² simplifies to x^(2/5). Divide the two expressions by subtracting the exponents, resulting in x^(3/5) as the final answer.
Step-by-step explanation:
To simplify ³√x³ ⁵√x², we need to simplify the radicals separately and then divide the simplified expressions. Let's start with the first radical, ³√x³. To simplify this, we can rewrite it as (x³)^(1/3), which is the same as taking the cube root of x³. The cube root of x³ is simply x. Now, let's simplify the second radical, ⁵√x². This can be rewritten as (x²)^(1/5), which is the same as taking the fifth root of x². The fifth root of x² is x^(2/5). Now we can divide the two simplified expressions, x / x^(2/5). To divide, we subtract the exponents, which would be 1 - (2/5). Simplifying further, we have x^(3/5) as the final answer.
The cube root of x cubed (³√x³) can be rewritten as x to the power of 1 (x¹), and the fifth root of x squared (⁵√x²) can be rewritten as x to the power of 2/5 (x²/⁵). When dividing powers with the same base, we subtract the exponents, yielding an answer in 200 words to the expression: x¹ - x²/⁵ = x¹ - (2/5) = x⁵/⁵ - (2/5) = x³/⁵, which is x raised to the three-fifths power.