Math Problem Statement

Write (6x^5y^3)^2 / (3x^2y^7 × 4xy^-3) in the form ax^by^c where a, b, and c are integers.

Solution

The question provided asks to simplify the expression:

(6x5y3)23x2y7×4xy3\frac{(6x^5y^3)^2}{3x^2y^7 \times 4xy^{-3}}

in the form axbycax^by^c, where aa, bb, and cc are integers.

Step-by-step solution:

  1. Expand the numerator: (6x5y3)2=62×(x5)2×(y3)2=36x10y6(6x^5y^3)^2 = 6^2 \times (x^5)^2 \times (y^3)^2 = 36x^{10}y^6

  2. Expand the denominator: 3x2y7×4xy3=3×4×x2×x1×y7×y3=12x3y43x^2y^7 \times 4xy^{-3} = 3 \times 4 \times x^2 \times x^1 \times y^7 \times y^{-3} = 12x^3y^4

  3. Divide the terms: 36x10y612x3y4=3612×x10x3×y6y4\frac{36x^{10}y^6}{12x^3y^4} = \frac{36}{12} \times \frac{x^{10}}{x^3} \times \frac{y^6}{y^4} Simplifying each factor: 3612=3,x10x3=x103=x7,y6y4=y64=y2\frac{36}{12} = 3, \quad \frac{x^{10}}{x^3} = x^{10-3} = x^7, \quad \frac{y^6}{y^4} = y^{6-4} = y^2

  4. Final result: 3x7y23x^7y^2

Thus, the expression is simplified to 3x7y23x^7y^2.


Would you like further details or explanations on any step? Here are some related questions:

  1. How do you expand exponents like (x5)2(x^5)^2?
  2. What are the rules for simplifying fractions with variables?
  3. How do you combine powers when multiplying terms with the same base?
  4. Why does dividing powers result in subtraction of exponents?
  5. What does the expression axbycax^by^c represent mathematically?

Tip: When simplifying algebraic expressions, always handle each variable separately using exponent rules.

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Math Problem Analysis

Mathematical Concepts

Algebra
Exponent Rules
Simplifying Expressions

Formulas

Power of a Power: (x^m)^n = x^(m*n)
Multiplying Powers: x^m * x^n = x^(m+n)
Dividing Powers: x^m / x^n = x^(m-n)

Theorems

Laws of Exponents

Suitable Grade Level

Grades 9-12