10 Questions
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.
like
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.
common
Expanding ______ involves multiplying out expressions involving products of binomials or polynomials.
products
To solve an equation, we can add or subtract the same value to ______ sides of the equation.
both
The ______ rule states that a^m * a^n = a^(m + n).
product
Simplifying ______ involves rewriting them in a simpler form, e.g., √(4x) = 2√x.
surds
Rationalizing ______ involves removing surds from denominators by multiplying by the conjugate.
denominators
Mastering algebraic manipulation skills is essential for solving ______ equations and inequalities.
algebraic
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
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.
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