Problem 32 - Beginning and Intermediate Algebra

Question

Use the distributive property to write each expression without parentheses Then simplify the result. See Example 4. \(7(a+b)\)

Step-by-Step Solution

Answer

The expression simplifies to \(7a + 7b\).

Step 1: Apply the distributive property

The distributive property states that you can distribute a factor over addition inside the parentheses: \[ a(b+c) = ab + ac \] In this case, distribute the 7 to both \(a\) and \(b\). This results in: \[ 7(a+b) = 7 \cdot a + 7 \cdot b \]

Step 2: Simplify the expression

Now carry out the multiplication for each term:\[ 7 \cdot a = 7a \]\[ 7 \cdot b = 7b \]Combine the two terms to get the simplified expression:\[ 7a + 7b \]

Key Concepts

simplifying expressions

In algebra, simplifying expressions involves making an equation as simple as possible by performing all possible operations and combining like terms. The objective is to rewrite the expression in a form that's easier to understand and work with. In the case of the distributive property, simplifying often involves expanding expressions and removing parentheses.When you apply simplification to the equation given, like in the expression \(7(a+b)\), you start by expanding it using the distributive property. This means you multiply each term inside the parentheses by the number outside, which in this case is 7. After you do this, you proceed with carrying out any arithmetic operations involved, such as the multiplication itself. Finally, after expanding using distribution and performing any remaining operations, you double-check for any like terms that can be combined.Understanding simplification is key because it allows one to handle more complex algebraic expressions easily, turning them into something more manageable.

algebraic expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations. Algebraic expressions can be as simple as a single term, like \(4x\), or a complex mix of multiple terms, like \(3x^2 + 2xy - y\).The power of algebraic expressions lies in their ability to represent real-world problems and unknown values mathematically. Each term in an algebraic expression is a product or a quotient, which can include variables raised to a power or constants.For the expression \(7(a+b)\), the expression initially includes a combination of a constant 7 and the algebraic sum \(a+b\). Applying the distributive property expands this into a more detailed expression without the parentheses, resulting in \(7a + 7b\). Each part of the original expression tells us something about the operation being represented, and although they seem symbolic, these notations are practical and handy for solving problems efficiently.

multiplication in algebra

Multiplication in algebra involves applying arithmetic rules to variables and constants within an expression. This means you use the same fundamental operations that you do with numbers, but you also need to pay attention to exponent rules and variable coefficients. The distributive property is a cornerstone process in algebra when multiplying. It allows you to simplify expressions with multiplication across addition or subtraction, as seen in \(a(b+c) = ab + ac\). In our example \(7(a+b)\), multiplication happens between the constant 7 and every term inside the parentheses: \(a\) and \(b\). Each term of \(a\) and \(b\) receives the 7 through the operation \(7 \cdot a\) and \(7 \cdot b\), resulting in \(7a + 7b\).Remember, successful multiplication in algebra involves keeping all terms aligned, recognizing when to apply distribution, and ensuring the rules of operations are correctly followed. Familiarity with these steps enhances your ability to manage increasingly complicated equations and expressions efficiently.