Math Problem Statement

sqrt((x^(2/3)+y^(2/3))/x^(2/3

Solution

It seems that you are working with an expression involving square roots and exponents. I assume your expression is:

x2/3+y2/3x2/3\sqrt{\frac{x^{2/3} + y^{2/3}}{x^{2/3}}}

Let's simplify this step by step.

  1. First, split the terms in the numerator: x2/3x2/3+y2/3x2/3\sqrt{\frac{x^{2/3}}{x^{2/3}} + \frac{y^{2/3}}{x^{2/3}}}

  2. Simplify x2/3x2/3=1\frac{x^{2/3}}{x^{2/3}} = 1, so the expression becomes: 1+y2/3x2/3\sqrt{1 + \frac{y^{2/3}}{x^{2/3}}}

  3. Recall that y2/3x2/3=(yx)2/3\frac{y^{2/3}}{x^{2/3}} = \left(\frac{y}{x}\right)^{2/3}, so the expression simplifies to: 1+(yx)2/3\sqrt{1 + \left(\frac{y}{x}\right)^{2/3}}

Thus, the simplified form of the expression is:

1+(yx)2/3\sqrt{1 + \left(\frac{y}{x}\right)^{2/3}}

Let me know if you want further explanation or additional details.


Here are five related questions you might want to consider:

  1. How do you simplify radicals involving fractional exponents?
  2. What are the properties of exponents that help simplify expressions like these?
  3. How would this expression change if the powers of xx and yy were different?
  4. Can you graph the function 1+(yx)2/3\sqrt{1 + \left(\frac{y}{x}\right)^{2/3}}?
  5. How does the behavior of the function change as x0x \to 0?

Tip: When dealing with expressions involving exponents and radicals, always try to rewrite them in a simpler form using properties of exponents for easier manipulation.

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Math Problem Analysis

Mathematical Concepts

Algebra
Exponents
Radicals

Formulas

Simplification of exponents: a^(m/n)
Properties of radicals: sqrt(a/b) = sqrt(a) / sqrt(b)

Theorems

Exponent Rules
Properties of Radicals

Suitable Grade Level

Grades 9-12