Algebraic Simplification

In the following examples and problems, the term "simplify" indicates to eliminate compound fractions, factor as much as possible, put terms over a common denominator when feasible, and avoid negative exponents.

Ex 1 Simplify the expression

$$x^2(4(x-2)^3)+2x(x-2)^4$$

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Sol

$$x^2(4(x-2)^3)+2x(x-2)^4=2x(x-2)^3[2x+(x-2)]=2x(x-2)^3[3x-2]$$

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Ex 2 Simplify the expression

$$\frac{(x^2+3)^2(6)-6x(2)(x^2+3)(2x)}{(x^2+3)^4}$$

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Sol

$$\frac{(x^2+3)^2(6)-6x(2)(x^2+3)(2x)}{(x^2+3)^4}$$

$$=\frac{6(x^2+3)[(x^2+3)-(2x)(2x)]}{(x^2+3)^4}$$

$$=\frac{6[x^2+3-4x^2]}{(x^2+3)^3}=\frac{6[3-3x^2]}{(x^2+3)^3}$$

$$=\frac{18(1-x^2)}{(x^2+3)^3}=\frac{18(1-x)(1+x)}{(x^2+3)^3}$$

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Pr A Simplify the expression $\frac{x^3-8}{x^2-4}$.

Pr B Simplify the expression

$$\frac{1/x^2-1/9}{x-3}$$

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Pr C Simplify the expression $\frac{\sqrt{x+h}-\sqrt{x}}{h}$by eliminating the radicals in the numerator.

Pr 1 Simplify the expression

$$5/3x^{2/3}-10/3x^{-1/3}$$

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Pr 2 Simplify the expression

$$x(1/2)(2x+3)^{-1/2}(2)+\sqrt{2x+3}$$

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Pr 3 Simplify the expression

$$t^2(1/2)(t-2)^{-1/2}+2t\sqrt{t-2}$$

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Pr 4 Simplify the expression

$$\sqrt{x}(2)(x-2)+(1/2)x^{-1/2}(x-2)^2$$

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Pr 5 Simplify the expression

$$x(1/3)(x^2+5)^{-2/3}(2x)+(x^2+5)^{1/3}$$

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Pr 6 Simplify the expression

$$\frac{\sqrt{25+x^2}-x(1/2)(25+x^2)^{-1/2}(2x)}{25+x^2}$$

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Pr 7 Simplify the expression

$$\frac{x(1/2)(x^2+1)^{-1/2}(2x)-\sqrt{x^2+1}}{x^2}$$

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Pr 8 Simplify the expression

$$\frac{(x^2+1)^2(-2x)-(1-x^2)(2)(x^2+1)(2x)}{(x^2+1)^4}$$

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