Question

DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the expression without parentheses. $$ -4(t-8) $$

Short answer

-4t + 32

Step-by-step solution

Apply the Distributive Property

To solve this, one needs to multiply -4 with each term inside the parentheses. Thus, the expression \( -4(t-8) \) becomes: \( -4 \cdot t + -4 \cdot -8 \)

Do the Multiplication

Now carry out the multiplication: \( -4t + (-4 \cdot -8) \)

Calculate the Value

Calculate the value, -4 multiplied by -8 is 32. Thus the expression simplifies to: \( -4t + 32 \)

Key concepts

Algebraic Expression

Understanding algebraic expressions is fundamental to grasping the basics of algebra. An algebraic expression is a combination of variables, numbers, and at least one arithmetic operation. In our example, -4(t-8), we have a variable t, numbers -4 and 8, and the operations are multiplication and subtraction. These expressions can represent numbers in terms of variables or 'unknowns', allowing us to describe and solve problems in a generalized way.

In algebra, the beauty lies in the ability to work with these symbols and numbers to simplify and solve equations, which often include algebraic expressions as part of their structure. The expression -4(t-8) is initially compact, but to understand and solve it, we spread it out into its components using algebraic principles like the distributive property.

Simplifying Expressions

Simplifying expressions is a way to make them easier to understand or work with. It involves reducing an expression to its most basic form without changing its value. The process may include combining like terms, using the distributive property, and performing arithmetic operations. In the case of the expression -4(t-8), we simplify it by applying the distributive property.

This means we distribute the -4 across the terms within the parentheses, multiplying it by each term. This step transforms the expression into a simpler, equivalent form, -4t + 32. Simplifying expressions is like cleaning a room; we are organizing and removing what's unnecessary so that the purpose of the room—or in our case, the expression—is clear and uncluttered.

Negative Numbers

Negative numbers are numbers with a value less than zero, denoted by a minus sign (-). They are essential when working with algebraic expressions and in mathematical operations such as subtraction and multiplication. They arise naturally when you calculate differences or describe values below a certain reference point, such as sea level or freezing temperature.

Understanding how to handle negative numbers is crucial when simplifying expressions. In our example, we see the negative number -4 multiplied by a negative number inside the parentheses, which, by the rules of arithmetic, gives us a positive result. Hence, -4 times -8 results in a positive 32. It's important to remember that multiplying two negative numbers always results in a positive number; this is because a negative times a negative is considered a double negation, effectively cancelling out the 'negativity' and producing a positive.