Problem 46 - Fundamentals of Mathematics

Question

For problems \(17-46\), find the value of each expression. $$ (a-7)^{2}-2(a-7)-2, \text { if } a=7 $$

Step-by-Step Solution

Answer

The value is -2.

Step 1: Substitute the value of 'a'

To solve the expression \((a-7)^{2}-2(a-7)-2\), we first substitute \(a=7\) into the expression. This gives us \((7-7)^{2}-2(7-7)-2\).

Step 2: Simplify inside the parenthesis

Calculate \(7-7\), which results in \(0\). So, the expression becomes \(0^{2}-2\times 0-2\).

Step 3: Evaluate powers and products

Calculate \(0^{2}\), which is \(0\), and \(2 \times 0\), which is also \(0\). The expression now reduces to \(0 - 0 - 2\).

Step 4: Perform final arithmetic operation

Subtract \(0\) and \(2\) from \(0\), giving \(-2\).

Key Concepts

Substitution

Substitution is the process of replacing a variable in an expression with a specific value. This is a fundamental concept in algebra, allowing us to evaluate expressions for given values. In this exercise, we replace the variable \( a \) with \( 7 \). By doing so, we transform the algebraic expression

• \((a-7)^2 - 2(a-7) - 2\)

into a numerical expression:

• \((7-7)^2 - 2(7-7) - 2\)
.

This step is crucial because it sets the stage for further simplification. The substitution process involves simple arithmetic and is often the first step in evaluating expressions.

Simplification

Simplification involves reducing an expression to its simplest form by performing operations inside parentheses first. In our exercise, this means calculating what is inside the parentheses. Replacing \(a\) with \(7\), we start with

• \( (7-7)^2 - 2(7-7) - 2 \).

We calculate \((7-7)\), which equals \(0\). By substituting this back into the expression, we get

• \( 0^2 - 2 imes 0 - 2 \).

This simplification process removes unnecessary complexity, making the expression easier to handle. Simplifying expressions allows us to clearly see the operations that need to be performed next.

Arithmetic Operations

Arithmetic operations in mathematics include addition, subtraction, multiplication, and division. These operations are key to solving expressions after simplification. After substituting and simplifying in this exercise, we have the expression

• \(0^2 - 2 \times 0 - 2\).

Firstly, we evaluate \(0^2\), which is \(0\). Next, we compute \(2 \times 0\), which also equals \(0\). This simplifies the expression to

• \(0 - 0 - 2\).

Finally, we carry out the subtraction operation, ending with

• \(-2\).

Each arithmetic operation transforms the expression closer to its final value. Mastery of these operations is essential for understanding and resolving math problems efficiently.