Problem 73
Question
Simplify the algebraic expressions for the following problems. $$ 4(x-8) $$
Step-by-Step Solution
Verified Answer
Answer: The simplified form of the expression is \(4x - 32\).
1Step 1: Distribute 4 to the terms within the parentheses
Multiply 4 by each term inside the parentheses:
$$
4(x-8) = 4 \cdot x - 4 \cdot 8
$$
2Step 2: Perform the multiplication
Perform the multiplication in the previous step:
$$
4 \cdot x - 4 \cdot 8 = 4x - 32
$$
3Step 3: Simplified expression
The simplified expression is:
$$
4x - 32
$$
Key Concepts
Distributive PropertyMultiplying VariablesAlgebraic SimplificationElementary Algebra
Distributive Property
Understanding the distributive property is fundamental in algebra, as it allows us to multiply a single term by each term within a parenthesis individually to 'distribute' the multiplication evenly. For instance, when we have an expression like \(4(x-8)\), the distributive property directs us to multiply the number 4 by each term inside the parentheses: \(x\) and \(-8\).
We apply it as follows: \(4 \times x\) and \(4 \times (-8)\). This property simplifies the process of dealing with parentheses, transforming them into easier terms to work with. It's valuable not only in arithmetic but also in solving more complex algebraic equations and simplifying expressions.
We apply it as follows: \(4 \times x\) and \(4 \times (-8)\). This property simplifies the process of dealing with parentheses, transforming them into easier terms to work with. It's valuable not only in arithmetic but also in solving more complex algebraic equations and simplifying expressions.
Multiplying Variables
When multiplying variables, keep in mind that a variable represents an unknown number, but the rules of multiplication stay the same. If a variable is preceded by a number (known as a coefficient), such as in \(4x\), you multiply the coefficient by the variable. However, if the variable does not have a specific number with it, assume the coefficient is 1.
In the case where variables are multiplied with other variables, such as \(x \times y\), the result is simply the variables written together, like \(xy\), which indicates that they're multiplied with each other. But remember, you can only combine like terms. So, if the variables are the same and have the same exponent, you can add their coefficients.
In the case where variables are multiplied with other variables, such as \(x \times y\), the result is simply the variables written together, like \(xy\), which indicates that they're multiplied with each other. But remember, you can only combine like terms. So, if the variables are the same and have the same exponent, you can add their coefficients.
Algebraic Simplification
Algebraic simplification involves reducing an expression to its simplest form while keeping its value equivalent to the original. This process includes combining like terms (terms with the same variable and exponent), applying the distributive property, and performing any addition or subtraction.
Take our example \(4x - 32\). It's already in its simplest form because there are no like terms to combine, and all the operations have been performed. To ensure that you have truly simplified an expression, check that the coefficients are whole numbers when possible, the variables and their respective exponents are in their basic form, and that all possible operations have been handled.
Take our example \(4x - 32\). It's already in its simplest form because there are no like terms to combine, and all the operations have been performed. To ensure that you have truly simplified an expression, check that the coefficients are whole numbers when possible, the variables and their respective exponents are in their basic form, and that all possible operations have been handled.
Elementary Algebra
Elementary algebra is the branch of mathematics that deals with variables in addition to numbers. It is where we start to learn how to work with unknowns and how to manipulate algebraic expressions and equations. The example \(4x - 32\) stems from elementary algebra and shows how we can take a problem with an unknown quantity, x, and simplify it using operations such as multiplication and subtraction.
This is just the starting point of algebra, which can later lead us to more complex problem-solving, including working with quadratic equations, functions, and even calculus. Mastering the foundational aspects of algebra, such as the distributive property and manipulating variables, is crucial for progress in any advanced mathematics.
This is just the starting point of algebra, which can later lead us to more complex problem-solving, including working with quadratic equations, functions, and even calculus. Mastering the foundational aspects of algebra, such as the distributive property and manipulating variables, is crucial for progress in any advanced mathematics.
Other exercises in this chapter
Problem 73
For the following problems, a term will be given followed by a group of its factors. List the coefficient of the given group of factors. $$ 9 a^{2} b^{5} ; 3 a
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(Section 4.6) Find the product. \((5 m-3)(2 m+3)\).
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For the following problems, simplify each of the algebraic expressions. $$ x y(3 x y+2 x-5 y)-2 x^{2} y^{2}-5 x^{2} y+4 x y^{2} $$
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For the following problems, perform the multiplications and combine any like terms. $$ (x+1)(x+7) $$
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