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Chapter 3: Problem 36

Simplify the expression. $$10(-10 x+1)$$

Short Answer

Expert verified
The simplified expression is \(-100x + 10\).

Step by step solution

01

Identify the expression

The given expression is \(10(-10x + 1)\). This consists of the number 10 being multiplied by the expression \(-10x + 1\).
02

Apply the distributive property

To simplify, apply the distributive property of multiplication over addition or subtraction. Multiply 10 by each term inside the parentheses: \(10 \times (-10x)\) and \(10 \times 1\).
03

Multiply each term

Perform the multiplications. For \(10 \times (-10x)\), we get \(-100x\). For \(10 \times 1\), we get 10.
04

Combine the results

The simplified expression is obtained by combining the results from the previous step. Thus, the expression simplifies to \(-100x + 10\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a fundamental principle in algebra that helps to simplify expressions. It allows you to multiply a single term and two or more terms inside a set of parentheses. Do you remember the phrase "distribute the wealth"? Here, the wealth is the number outside the parentheses!
For the expression \(10(-10x + 1)\), we use the distributive property to multiply 10 by each term in the parentheses separately. It looks like this:
  • First, multiply 10 by \(-10x\)
  • Then, multiply 10 by 1
By doing this, we break down complex expressions into simpler parts, making them easier to handle and solve.
Multiplication of Terms
After distributing, we arrive at the multiplication of individual terms. In algebra, multiplying terms involves multiplying both their numerical and variable components.
For example, in the exercise, you multiply 10 by \(-10x\) and 10 by 1. Let's see how these multiplications work:
  • For \(10 \times (-10x)\), multiply the numbers 10 and -10 to get -100, and keep the \(x\) with it, resulting in \(-100x\).
  • For \(10 \times 1\), it鈥檚 straightforward multiplication, where you simply get 10.
Each multiplication step is vital in ensuring that no term is left behind or improperly calculated. Thoroughly understanding this will help you tackle any algebraic expression with ease.
Algebraic Expressions
Algebraic expressions are a combination of constants, variables, and arithmetic operations like addition, subtraction, multiplication, and division. Understanding how to manipulate these expressions is crucial in solving various mathematical problems.

In this exercise, the expression \(10(-10x + 1)\) is an algebraic expression involving a constant (10), a variable (\(x\)), and operations (multiplication and addition). Simplifying algebraic expressions requires:
  • Applying properties of algebra like the distributive property
  • Performing arithmetic operations like multiplication
  • Combining like terms when applicable
By breaking down expressions into simpler components, algebra becomes less intimidating and more approachable. Remember, practice makes perfect. The more you work with expressions, the more comfortable you'll become.

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