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Chapter 1: Problem 30

Evaluate each expression. $$ (-0.8)^{2} $$

Short Answer

Expert verified
The expression evaluates to 0.64.

Step by step solution

01

Identify the Base and Exponent

In the expression \((-0.8)^2\), the base is \(-0.8\) and the exponent is 2. This means we need to multiply \(-0.8\) by itself.
02

Perform the Multiplication

Multiply \(-0.8\) by \(-0.8\). When multiplying two negative numbers, the result is positive: \((-0.8) \times (-0.8) = 0.64\).
03

Confirm the Result

Ensure that the multiplication follows the rules for negative numbers, confirming that the negative signs cancel out, resulting in a positive product.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Negative Numbers
Negative numbers are values less than zero and are represented with a minus sign (-) in front of them. They are used in mathematics to represent opposite quantities, like owing money or temperatures below zero. When dealing with negative numbers in mathematical operations, especially multiplication, it's crucial to understand how they interact with each other.
A negative number multiplied by another negative number results in a positive product. This is because the two negative signs cancel each other out. For example, multiplying (-0.8) by (-0.8), we get a positive result of 0.64.
  • A single negative sign remains negative.
  • An even number of negative signs turns positive.
  • An odd number of negative signs remains negative.
Understanding these simple rules can simplify working with negative numbers in any expression.
Rules of Multiplication
Multiplication involves combining two numbers to produce a sum total value. This rule applies succinctly even to special cases like negative numbers and zero. Understanding the basic rules of multiplication provides a strong foundation for solving various mathematical expressions.
  • The product of two positive numbers is positive.
  • The product of two negative numbers is positive, as mentioned before.
  • The product of a positive number and a negative number is negative.
  • Any number multiplied by zero equals zero.
Knowing these rules helps us navigate more complex expressions, including exponentiation with negative bases. As seen in the example, squaring (-0.8) requires us te multiply it by itself, confirming that negative multiplied by negative is positve, thus (-0.8) (-0.8) equals 0.64.
Working with Mathematical Expressions
Mathematical expressions are combinations of numbers, operations (like addition and multiplication), and sometimes variables. In our example, we worked with an expression involving a base and an exponent: (-0.8)^2.
Breaking down this expression involved:
  • Identifying the base: (-0.8).
  • Recognizing the exponent: 2.
  • Applying the multiplication rules to solve the power (exponent) part of the expression.
To solve, we applied the rules of multiplication to handle the base raised to a power. This involved repetitive multiplication of the base by itself, as shown: (-0.8) (-0.8) = 0.64.
Conclusively, understanding the components of a mathematical expression is vital as it lays down a framework to approach and solve mathematical problems methodically.

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