Problem 38 - Prealgebra

Question

Perform the following operations according to the rule for order of operations. $$(9.6-0.5)(9.6+0.5)$$

Step-by-Step Solution

Answer

The result is 91.91.

Step 1: Identify the Expression

The expression given is \((9.6 - 0.5)(9.6 + 0.5)\). This is a product of two binomials.

Step 2: Recognize the Formula

Notice that the expression is in the form \((a - b)(a + b)\), which is the difference of squares formula: \((a - b)(a + b) = a^2 - b^2\).

Step 3: Assign Values

For our expression, let \(a = 9.6\) and \(b = 0.5\). We will use these values in the formula.

Step 4: Apply the Formula

Substitute \(a\) and \(b\) into the difference of squares formula: \((9.6)^2 - (0.5)^2\).

Step 5: Calculate Squares

Calculate \(9.6^2\) and \(0.5^2\):- \(9.6^2 = 92.16\)- \(0.5^2 = 0.25\).

Step 6: Subtract Values

Now subtract the squared values: \(92.16 - 0.25 = 91.91\).

Step 7: Arrive at the Final Answer

The final result of the expression \((9.6 - 0.5)(9.6 + 0.5)\) is 91.91.

Key Concepts

Difference of Squares

One key concept utilized in the exercise is the difference of squares, a special algebraic identity that simplifies the product of two specific binomials. When you have a pair of binomials in the form

• \((a - b)(a + b)\)

.it can be transformed into

• \(a^2 - b^2\).

This formula arises from the fact that the middle terms cancel each other out:

• \((a + b)(a - b) = a^2 + ab - ab - b^2\),

after canceling \(ab\), you're left with the simplified form.
This powerful formula allows you to quickly compute expressions that might otherwise require more steps. It is especially useful in algebra for simplifying calculations and solving equations.

Binomial Multiplication

Binomial multiplication refers to the process of multiplying two terms that each consist of two parts, valuable in algebra when dealing with expressions like \((x + y)(x - y)\). To correctly multiply two binomials, you should perform a method that multiplies each term in the first binomial by each term in the second:

• Multiply the first terms: \(a \times a = a^2\).
• Multiply the outer terms: \(a \times b = ab\).
• Multiply the inner terms: \(-b \times a = -ab\).
• Multiply the last terms: \(-b \times b = -b^2\).

When put together, this process is often remembered by the acronym "FOIL" (First, Outer, Inner, Last). In the case of our expression \((9.6 - 0.5)(9.6 + 0.5)\), it's evident that the outer and inner products cancel out, emphasizing the simplicity and effectiveness of using the difference of squares formula.

Mathematical Expressions

Understanding and simplifying mathematical expressions is a crucial skill in mathematics, allowing you to manipulate and solve complex problems efficiently. A mathematical expression can contain numbers, variables, and operators designed to represent a value or an equation.

Key components include:

• Numbers: These represent constant values.
• Variables: Symbols like \(x\) or \(y\) that can take on different values.
• Operators: Symbols like addition (+), subtraction (-), multiplication (×), and division (÷) that indicate operations to perform.

In our specific example, the expression \((9.6 - 0.5)(9.6 + 0.5)\) involves both constants and operators. Breaking this down into simpler forms or using identities like the difference of squares can make the problem more efficient to solve.
When working with expressions, following the order of operations (P-E-M-D-A-S: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) ensures that calculations are done correctly.