Problem 36 - Fundamentals of Mathematics

Question

Find the value of each expression. $$-8(5 m+8 n), \text { if } m=0 \text { and } n=-1$$

Step-by-Step Solution

Answer

The value of the expression is 64.

Step 1: Substitute the Values of Variables

Given the expression \(-8(5m + 8n)\). First, substitute \(m = 0\) and \(n = -1\) into the expression. This gives:\[-8(5(0) + 8(-1))\]

Step 2: Simplify Inside the Parentheses

Simplify the expression inside the parentheses. Calculate \(5 \times 0\) and \(8 \times -1\):\[5 \times 0 = 0 \8 \times -1 = -8\]Substitute these into the expression:\[-8(0 - 8)\]

Step 3: Further Simplify Inside the Parentheses

Now simplify the expression inside the parentheses:\[0 - 8 = -8\]The expression becomes:\[-8(-8)\]

Step 4: Multiply the Outer Coefficient

Finally, multiply \(-8\) by \(-8\):\[-8 \times -8 = 64\]

Key Concepts

Substitution

Substitution is all about replacing variables in an algebraic expression with specific values. In our exercise, we're given the expression \(-8(5m + 8n)\) and the specific values \(m = 0\) and \(n = -1\). The goal is to find the value of this expression by substituting these values in place of the variables.

• Why do we substitute? - It helps convert the expression from one dealing with variables to one that contains only numbers, making it easier to simplify and solve.
• How do we substitute? - Wherever you see \(m\) in the expression, replace it with \(0\), and wherever you see \(n\), replace it with \(-1\). So, \(-8(5m + 8n)\) becomes \(-8(5 \times 0 + 8 \times -1)\).

Remember, substitution is a key step in solving algebraic problems, making them much more manageable.

Simplification

Simplification is the process of making an expression easier to work with by reducing it to its simplest form. In our example, we started with the expression \(-8(5 \times 0 + 8 \times -1)\). After performing substitution, simplification allows us to break it down into more simple and manageable terms.

• First, calculate each term: \(5 \times 0 = 0\) and \(8 \times -1 = -8\).
• Then, you'll have \(-8(0 - 8)\).
• Next, simplify within the parentheses: \(0 - 8 = -8\).

Simplification doesn't change the value of an expression; it just makes it easier to handle. It often requires performing arithmetic operations like multiplication and addition to combine terms or reduce expressions.

Multiplication of Integers

Multiplication of integers is a fundamental arithmetic operation used when simplifying expressions. When you multiply integers, you follow specific rules based on the signs of the numbers involved.

• Same sign: Multiplying two positive or two negative integers results in a positive product. For instance, \(-8 \times -8\) gives us \(64\).
• Different signs: Multiplying a positive and a negative integer results in a negative product. For example, \(-2 \times 3 = -6\).

In our exercise, after simplification, we had \(-8(-8)\). Applying the rule above, multiplying two negative numbers yields a positive result, which is \(64\). Understanding these rules ensures that your solutions are accurate when dealing with multiplication in algebraic expressions.