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nameVoid
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How Do You Simplify Complex Algebraic Expressions?
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SUMMARY
The discussion focuses on simplifying the complex algebraic expression {(\sqrt{a} + 1)^2 - {a-\sqrt {ax} \over \sqrt a - \sqrt x} \over (\sqrt a + 1)^3 - a \sqrt a + 2}. The simplification process involves expanding and rearranging terms, ultimately leading to the simplified result of {1 \over 3}. Key techniques include grouping like terms and canceling common factors to achieve the final expression.
PREREQUISITES- Understanding of algebraic expressions and simplification techniques
- Familiarity with square roots and their properties
- Knowledge of grouping and factoring in algebra
- Ability to manipulate fractions and complex fractions
- Study advanced algebraic manipulation techniques
- Learn about polynomial long division and its applications
- Explore the use of algebraic identities in simplification
- Practice simplifying complex expressions using symbolic computation tools like Wolfram Alpha
Students, educators, and anyone looking to enhance their skills in algebraic simplification and expression manipulation.
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whybother
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So you are trying to simplify the expression, I take it?
[tex]{(\sqrt{a} + 1)^2 - {a-\sqrt {ax} \over \sqrt a - \sqrt x} \over (\sqrt a + 1)^3 - a \sqrt a + 2}[/tex]
= [tex]{(a + 2 \sqrt a + 1)(\sqrt a - \sqrt x) - (a - \sqrt {ax}) \over (\sqrt a - \sqrt x)((\sqrt a + 1)^3 - a \sqrt a + 2)}[/tex]
= [tex]{a \sqrt a - a \sqrt x +2a - 2 \sqrt a \sqrt x + \sqrt a - \sqrt x - a + \sqrt{ax} \over (\sqrt a - \sqrt x)(a^{3 \over 2} + 3a + 3 \sqrt a + 1 - a^{3 \over 2} + 2)}[/tex]
And then we group like terms and can cancel some things out
=[tex]{a \sqrt a + a + \sqrt a - a \sqrt x - \sqrt{ax} - \sqrt x \over 3(a \sqrt a + a + \sqrt a - a \sqrt x - \sqrt a \sqrt x - \sqrt x}[/tex]
= [tex]{1 \over 3}[/tex]
[tex]{(\sqrt{a} + 1)^2 - {a-\sqrt {ax} \over \sqrt a - \sqrt x} \over (\sqrt a + 1)^3 - a \sqrt a + 2}[/tex]
= [tex]{(a + 2 \sqrt a + 1)(\sqrt a - \sqrt x) - (a - \sqrt {ax}) \over (\sqrt a - \sqrt x)((\sqrt a + 1)^3 - a \sqrt a + 2)}[/tex]
= [tex]{a \sqrt a - a \sqrt x +2a - 2 \sqrt a \sqrt x + \sqrt a - \sqrt x - a + \sqrt{ax} \over (\sqrt a - \sqrt x)(a^{3 \over 2} + 3a + 3 \sqrt a + 1 - a^{3 \over 2} + 2)}[/tex]
And then we group like terms and can cancel some things out
=[tex]{a \sqrt a + a + \sqrt a - a \sqrt x - \sqrt{ax} - \sqrt x \over 3(a \sqrt a + a + \sqrt a - a \sqrt x - \sqrt a \sqrt x - \sqrt x}[/tex]
= [tex]{1 \over 3}[/tex]
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