Math Problem Statement

Simplify the expression: (x^3 * y^(1/2)) / (5 * 3 * x^2 * y^2 * y^(-1/3))

Solution

The given expression is:

x3y1/253x2y2y1/3\frac{\frac{x^3 y^{1/2}}{5 \cdot 3}}{x^2 y^2 \cdot y^{-1/3}}

Step-by-step simplification:

  1. Simplify the numerator:
    The numerator is x3y1/215\frac{x^3 y^{1/2}}{15} (since 5×3=155 \times 3 = 15).

  2. Simplify the denominator:
    In the denominator, we have: x2y2y1/3=x2y21/3=x2y5/3x^2 y^2 \cdot y^{-1/3} = x^2 y^{2 - 1/3} = x^2 y^{5/3}

  3. Rewrite the expression: Now, the expression becomes: x3y1/215x2y5/3\frac{\frac{x^3 y^{1/2}}{15}}{x^2 y^{5/3}}

  4. Simplify the fraction: =x3y1/215x2y5/3= \frac{x^3 y^{1/2}}{15 \cdot x^2 y^{5/3}}

  5. Simplify the powers of xx and yy: For xx, subtract the exponents: x32=x1x^{3 - 2} = x^1 For yy, subtract the exponents: y1/25/3=y36106=y76y^{1/2 - 5/3} = y^{\frac{3}{6} - \frac{10}{6}} = y^{-\frac{7}{6}}

    So the expression becomes: x15y7/6\frac{x}{15 y^{7/6}}

Final simplified expression:

x15y7/6\frac{x}{15 y^{7/6}}

Would you like further clarification or have additional questions? Here are some related questions:

  1. How do you simplify expressions with fractional exponents?
  2. What is the process for dividing variables with exponents?
  3. How do you combine like terms in algebraic fractions?
  4. Can you explain how negative exponents affect the simplification process?
  5. What rules govern the simplification of rational expressions?

Tip: Always double-check your work by breaking down exponents step by step to avoid mistakes!

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Math Problem Analysis

Mathematical Concepts

Algebra
Rational Expressions
Exponent Rules

Formulas

Product of exponents: a^m * a^n = a^(m + n)
Division of exponents: a^m / a^n = a^(m - n)
Simplifying fractional exponents

Theorems

Laws of Exponents
Properties of Rational Expressions

Suitable Grade Level

Grades 9-10