Problem 40 - Prealgebra

Question

Simplify each expression. $$10 \cdot(z \cdot 9)$$

Step-by-Step Solution

Answer

The simplified expression is \(90z\).

Step 1: Apply the Commutative Property of Multiplication

When multiplying numbers and variables, we can rearrange them as needed due to the commutative property of multiplication. This property tells us that for any numbers or variables \(a\) and \(b\), \(a \cdot b = b \cdot a\). Thus, we can rewrite the expression \(10 \cdot (z \cdot 9)\) as \((10 \cdot 9) \cdot z\) or \(z \cdot (10 \cdot 9)\).

Step 2: Simplify the Numerical Multiplication

Now, calculate the multiplication of the constant numbers. We have \(10 \cdot 9\). Multiply these numbers: \[ 10 \times 9 = 90 \].

Step 3: Rewrite the Expression

Substitute the result from Step 2 back into the expression. We now have \(90 \cdot z\), or equivalently, \(90z\). This is the simplified form of the original expression.

Key Concepts

Simplifying Expressions

When we talk about simplifying expressions in algebra, we're really discussing a way to make these expressions easier to work with. Think of it as cleaning up or streamlining the math. Instead of dealing with a cumbersome equation with various numbers and variables, the goal is to get it into the simplest form possible.

Take our example, where we have the expression \(10 \cdot (z \cdot 9)\). Initially, this looks a bit complex due to the brackets and multiple multiplication terms. The first step to simplification is applying the properties of multiplication, such as the commutative property, to rearrange and combine those numbers for a cleaner expression.

By rewriting it as \((10 \cdot 9) \cdot z\), and then simplifying \(10 \cdot 9\) to \(90\), we've just made the expression simpler: \(90z\). Whenever you're faced with expressions like these, remember that the goal is simplification by eliminating unnecessary complexity. It makes solving and understanding the expression much more straightforward.

Numerical Multiplication

Numerical multiplication is a fundamental concept in mathematics that involves multiplying two or more numbers together to get a single product. This operation is not only easy to grasp but also crucial for simplifying algebraic expressions.

In our specific example, the expression \(10 \cdot (z \cdot 9)\), requires multiplying the numbers \(10\) and \(9\). Numerical multiplication follows the basic multiplication rules that probably everyone knows — the repeated addition of a number. Here, we simply use the fact that \(10\) multiplied by \(9\) is \(90\).

This step is essential because once the numerical values have been combined into one product, it becomes effortless to work with any variables, like \(z\) in our expression. So, numerical multiplication is an indispensable tool in simplifying expressions and making algebra more approachable.

Variables in Algebra

Let's explore the role of variables in algebra. In simple terms, variables are symbols, usually letters, representing numbers and values that don't change, or that we don't know yet. Using variables allows us to generalize mathematical concepts and express equations that can solve many different problems.

For instance, in our expression \(90z\), the letter \(z\) is the variable. It signifies an unknown number or quantity that interacts with the numerical part of our expression, which is the \(90\). The presence of a variable allows this expression to represent an infinite number of possibilities, depending on the value \(z\) takes.

Understanding variables is key because they lay the groundwork for solving equations and problems within algebra. They enable us to form equations based on real-world problems, generalize solutions, and even set the stage for more advanced topics like functions and calculus. When simplifying expressions, always remember the variable's role in maintaining the relationships and balancing equations.