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Questions and Answers
Algebraic ______ involves rearranging and simplifying algebraic expressions to make them more manageable and solvable.
Algebraic ______ involves rearranging and simplifying algebraic expressions to make them more manageable and solvable.
manipulation
Combining ______ terms is a technique used in simplifying algebraic expressions.
Combining ______ terms is a technique used in simplifying algebraic expressions.
like
To simplify an expression, we can ______ out terms that are equal but opposite in sign.
To simplify an expression, we can ______ out terms that are equal but opposite in sign.
cancel
Factoring out ______ factors is a technique used in simplifying algebraic expressions.
Factoring out ______ factors is a technique used in simplifying algebraic expressions.
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Expanding ______ involves multiplying out expressions involving products of binomials or polynomials.
Expanding ______ involves multiplying out expressions involving products of binomials or polynomials.
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To solve an equation, we can add or subtract the same value to ______ sides of the equation.
To solve an equation, we can add or subtract the same value to ______ sides of the equation.
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The ______ rule states that a^m * a^n = a^(m + n).
The ______ rule states that a^m * a^n = a^(m + n).
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Simplifying ______ involves rewriting them in a simpler form, e.g., √(4x) = 2√x.
Simplifying ______ involves rewriting them in a simpler form, e.g., √(4x) = 2√x.
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Rationalizing ______ involves removing surds from denominators by multiplying by the conjugate.
Rationalizing ______ involves removing surds from denominators by multiplying by the conjugate.
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Mastering algebraic manipulation skills is essential for solving ______ equations and inequalities.
Mastering algebraic manipulation skills is essential for solving ______ equations and inequalities.
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Study Notes
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying algebraic expressions to make them more manageable and solvable. Here are some key concepts and techniques:
Simplifying Expressions
- Combining like terms: Combine terms with the same variable(s) and coefficient(s).
- Canceling out terms: Remove terms that are equal but opposite in sign.
- Factoring out common factors: Extract common factors from terms to simplify the expression.
Expanding and Factorizing
- Expanding products: Multiply out expressions involving products of binomials or polynomials.
- Factorizing quadratic expressions: Express quadratic expressions in the form
a(x + b)(x + c).
Solving Equations and Inequalities
- Adding or subtracting the same value to both sides: Isolate the variable by adding or subtracting the same value to both sides of the equation.
- Multiplying or dividing both sides by the same non-zero value: Isolate the variable by multiplying or dividing both sides of the equation.
- Using inverse operations: Use inverse operations to solve equations, such as adding the opposite to get rid of addition, or multiplying by the reciprocal to get rid of multiplication.
Rules of Indices
- Product rule:
a^m * a^n = a^(m + n) - Quotient rule:
a^m / a^n = a^(m - n) - Power rule:
(a^m)^n = a^(m * n) - Zero index rule:
a^0 = 1
Surds and Rationalizing
- Simplifying surds: Simplify surds by rewriting them in a simpler form, e.g.,
√(4x) = 2√x - Rationalizing denominators: Remove surds from denominators by multiplying by the conjugate, e.g.,
1 / √2 = √2 / 2
These are some of the key concepts and techniques used in algebraic manipulation. Mastering these skills is essential for solving algebraic equations and inequalities.
Algebraic Manipulation
Simplifying Expressions
- Combine like terms by combining terms with the same variable(s) and coefficient(s)
- Cancel out terms that are equal but opposite in sign
- Factor out common factors from terms to simplify the expression
Expanding and Factorizing
- Expand products by multiplying out expressions involving products of binomials or polynomials
- Factorize quadratic expressions in the form
a(x + b)(x + c)
Solving Equations and Inequalities
- Isolate the variable by adding or subtracting the same value to both sides of the equation
- Isolate the variable by multiplying or dividing both sides of the equation by the same non-zero value
- Use inverse operations to solve equations, such as adding the opposite to get rid of addition, or multiplying by the reciprocal to get rid of multiplication
Rules of Indices
-
a^m * a^n = a^(m + n)(product rule) -
a^m / a^n = a^(m - n)(quotient rule) -
(a^m)^n = a^(m * n)(power rule) -
a^0 = 1(zero index rule)
Surds and Rationalizing
- Simplify surds by rewriting them in a simpler form, e.g.,
√(4x) = 2√x - Rationalize denominators by removing surds from denominators by multiplying by the conjugate, e.g.,
1 / √2 = √2 / 2
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Description
Learn how to simplify algebraic expressions by combining like terms, canceling out terms, and factoring out common factors. Master the key concepts and techniques of algebraic manipulation.
