Problem 101
Question
Factor and simplify each algebraic expression. $$(4 x-1)^{\frac{1}{2}}-1(4 x-1)^{\frac{3}{2}}$$
Step-by-Step Solution
Verified Answer
The factored and simplified form of the given expression is \((4x - 1)^{\frac{1}{2}}[-4x + 2]\)
1Step 1: Recognizing the Common Factor
Recognizing the common factor in the given expression is the first step. In our case, \((4x - 1)^{\frac{1}{2}}\) is our common factor between the two terms.
2Step 2: Factoring Out the Common Factor
Factor out the common factor. If we factor out \((4x - 1)^{\frac{1}{2}}\) from our expression, we get \[(4x - 1)^{\frac{1}{2}} [1 - (4x - 1)]\].
3Step 3: Simplify The Expression
Simplify the expression inside the bracket to obtain a simplified form. This gives us \[(4x - 1)^{\frac{1}{2}} [1 - (4x) + 1]\], which is \[(4x - 1)^{\frac{1}{2}}[-4x + 2]\]. Please note that the sign of 4x changed due to the subtraction operation.
Key Concepts
Understanding Common FactorsSimplification of Algebraic ExpressionsDiving into Algebraic Expressions
Understanding Common Factors
In the world of algebra, a common factor is a number or an expression that divides two or more terms completely without a remainder. Identifying a common factor is crucial for simplifying expressions, as it allows us to factor terms into simpler forms.
To find a common factor in algebraic expressions, we look for shared components in all terms. Consider two expressions, like \((4x - 1)^{\frac{1}{2}}\) and \((4x - 1)^{\frac{3}{2}}\). Here, both terms share the factor \((4x - 1)^{\frac{1}{2}}\). Recognizing this commonality helps to reduce the complexity of an expression.
By factoring out the common factor, the expression is rewritten in a form that groups what is shared, paving the way for further simplification.
To find a common factor in algebraic expressions, we look for shared components in all terms. Consider two expressions, like \((4x - 1)^{\frac{1}{2}}\) and \((4x - 1)^{\frac{3}{2}}\). Here, both terms share the factor \((4x - 1)^{\frac{1}{2}}\). Recognizing this commonality helps to reduce the complexity of an expression.
By factoring out the common factor, the expression is rewritten in a form that groups what is shared, paving the way for further simplification.
Simplification of Algebraic Expressions
Simplification is the process of making an algebraic expression as simple as possible, which often involves factoring, combining like terms, or eliminating unnecessary parts.
In algebra, once a common factor has been factored out, the next step is to simplify what remains. For instance, after factoring \((4x - 1)^{\frac{1}{2}}\) from \((4x - 1)^{\frac{1}{2}} [1 - (4x - 1)]\), we simplify the inner expression by distributing and performing operations like subtraction.
This example simplifies to \((4x - 1)^{\frac{1}{2}} [-4x + 2]\), where arithmetic operations inside the brackets are carried out to their simplest form. Through simplification, expressions become easier to understand and solve.
In algebra, once a common factor has been factored out, the next step is to simplify what remains. For instance, after factoring \((4x - 1)^{\frac{1}{2}}\) from \((4x - 1)^{\frac{1}{2}} [1 - (4x - 1)]\), we simplify the inner expression by distributing and performing operations like subtraction.
This example simplifies to \((4x - 1)^{\frac{1}{2}} [-4x + 2]\), where arithmetic operations inside the brackets are carried out to their simplest form. Through simplification, expressions become easier to understand and solve.
Diving into Algebraic Expressions
An algebraic expression is a mathematical phrase that can include numbers, variables, and operating symbols, representing a specific value or set of values.
Algebraic expressions are the building blocks of algebra and are manipulated through various processes such as addition, subtraction, multiplication, division, and exponentiation. When dealing with algebraic expressions, we often encounter tasks such as factoring and simplification.
In the expression \((4x - 1)^{\frac{1}{2}} - (4x - 1)^{\frac{3}{2}}\), different operations and manipulations need to be performed to arrive at a simpler form. This involves understanding how variables interact with coefficients and operations like exponentiation and factoring.
Understanding these concepts can aid in solving equations and manipulating expressions effectively.
Algebraic expressions are the building blocks of algebra and are manipulated through various processes such as addition, subtraction, multiplication, division, and exponentiation. When dealing with algebraic expressions, we often encounter tasks such as factoring and simplification.
In the expression \((4x - 1)^{\frac{1}{2}} - (4x - 1)^{\frac{3}{2}}\), different operations and manipulations need to be performed to arrive at a simpler form. This involves understanding how variables interact with coefficients and operations like exponentiation and factoring.
Understanding these concepts can aid in solving equations and manipulating expressions effectively.
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Problem 100
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