Question

Multiply or divide, as indicated. (x2 - y2)/(x - y)2 * (x2 - xy + y2)/(x2 - 2xy + y2) ÷ (x3 + y3)/(x - y)4

Accepted Answer

Start by rewriting the entire expression clearly, noting that division by a fraction is the same as multiplication by its reciprocal. The expression is:  
$$\frac{x^2$ - y^2$}{(x - y)^2$} 	imes \frac{x^2$ - xy + y^2$}{x^2$ - 2xy + y^2$} \div \frac{x^3$ + y^3$}{(x - y)^4$}$$  
which can be rewritten as  
$$\frac{x^2$ - y^2$}{(x - y)^2$} 	imes \frac{x^2$ - xy + y^2$}{x^2$ - 2xy + y^2$} 	imes \frac{(x - y)^4$}{x^3$ + y^3$}. $$Factor each polynomial where possible:  
- $$x^2$$ - $$y^2 is a $$difference of squares: $$x^2$$ - $$y^2 = (x - y)(x + y$$)$$  
- x^2$ - 2xy + y^2$ is a perfect square trinomial: x^2$ - 2xy + y^2$ = (x - y)^2$  
- x^3$ + y^3$ is a sum of cubes: x^3$ + y^3$ = (x + y)(x^2$ - xy + y^2$)$$  
Note that $$x^2 - xy$$ + $$y^2$$ does not factor further over the reals. Substitute the factored forms back into the expression:  
$$\frac{(x - y)(x + y)}{(x - y)^2$} 	imes \frac{x^2$ - xy + y^2$}{(x - y)^2$} 	imes \frac{(x - y)^4$}{(x + y)(x^2$ - xy + y^2$)}. $$Next, combine all numerators and denominators into a single fraction:  
Numerator: $$(x - y)(x + y)(x^2$ - xy + y^2$)(x - y)^4$  
Denominator: (x$$ - $$y)^2 (x$$ - $$y)^2 (x$$ + $$y)(x^2 - xy$$ + $$y^2$$)$$  
Simplify the powers of (x - y$$)$$ in numerator and denominator by adding and subtracting exponents. Cancel common factors in numerator and denominator:  
- Cancel (x + y$$)$$ terms  
- Cancel (x^2$ - xy + y^2$)$$ terms  
- Simplify powers of $$(x - y) by $$subtracting exponents in numerator and denominator  
After cancellation, write the simplified expression clearly.

Multiply or divide, as indicated. (x² - y²)/(x - y)² * (x² - xy + y²)/(x² - 2xy + y²) ÷ (x³ + y³)/(x - y)⁴

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Key Concepts

Factoring Polynomials

Operations with Rational Expressions

Simplifying Algebraic Expressions