Problem 71 - Prealgebra

Question

Simplify the given expression. \(9 .4-(-7.7)(1.2)^{2}\)

Step-by-Step Solution

Answer

The expression simplifies to 20.488.

Step 1: Simplify Powers

Start by simplifying the power part of the expression. Calculate \((1.2)^2\) which means multiplying 1.2 by itself: \(1.2 \times 1.2 = 1.44\). Substitute this back into the expression to get \(9.4 - (-7.7)(1.44)\).

Step 2: Multiply

Now multiply \(-7.7\) by \(1.44\). Compute \(-7.7 \times 1.44 = -11.088\). Substitute this result back into the expression: \(9.4 - (-11.088)\).

Step 3: Subtract Negative

Subtracting a negative number is the same as adding its positive. So, \(9.4 - (-11.088)\) becomes \(9.4 + 11.088\).

Step 4: Final Addition

Now add \(9.4\) to \(11.088\). Calculate \(9.4 + 11.088 = 20.488\). This is the simplified value of the expression.

Key Concepts

Simplifying Expressions

Simplifying expressions might seem like a small step, but it's a crucial one when solving mathematical problems. When you simplify an expression, you make it easier to handle, often reducing it to its most basic form. In our example, \(9.4 - (-7.7)(1.2)^2\), the process of simplification is about organizing the terms in a logical and simpler manner.

First, always follow the order of operations, commonly remembered as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). You initially deal with powers or exponents before moving onto multiplication and then addition/subtraction.

• Start with the exponent: calculate \((1.2)^2\).
• Follow with multiplication: \((-7.7)(1.44)\) simplifies after calculating the power.
• Conclude with the final addition or subtraction.

Understanding the flow of simplification helps in tackling more complex algebraic expressions later on.

Powers

Powers are a mathematical shorthand for repeated multiplication. The notation \((1.2)^2\) tells us to multiply 1.2 by itself, yielding \(1.44\). Powers make computations more compact and are frequently encountered across various math levels.

This concept, while simple, builds the foundation for understanding more advanced operations like roots and logarithms.

• To compute \((1.2)^2\), write it as \(1.2 \times 1.2\).
• The result \(1.44\) is then used in subsequent calculations.

When solving expressions, always compute powers before proceeding to other operations. This ensures accuracy and aligns with the correct order of operations that prevents mistakes in more complex expressions.

Negative Numbers

Negative numbers introduce an additional layer of complexity in equations. They represent values less than zero and can change the entire operation based on their placement. In our example, we encounter negative numbers in the term \(-7.7\), which interacts with the rest of the expression differently than positive numbers do.

Key aspects to remember about negative numbers include:

• Multiplying a negative by a positive yields a negative, as in \(-7.7 \times 1.44 = -11.088\).
• Subtracting a negative number is equivalent to adding its absolute value: \(9.4 - (-11.088) = 9.4 + 11.088\).

Handling negative numbers correctly can avoid sign errors and lead to correct solutions. Being cautious with operations involving negatives ensures that your final results are both accurate and reliable.