Problem 24 - Algebra A Combined Function

Question

Evaluate \(-0.8^{2}\)

Step-by-Step Solution

Answer

The value of \(-0.8^2\) is \(-0.64\).

Step 1: Understand the Expression

The expression \(-0.8^2\) looks like it involves a negative base being squared. It's important to distinguish that the negative sign is not within parentheses, meaning it does not apply to the base as a unit.

Step 2: Interpret the Negative Sign

The negative sign in front of the base indicates that we are dealing with the number \(-0.8\) as a negative value being raised to the power of 2. Thus, we interpret this as \(-1\times 0.8^2\).

Step 3: Square the Base

Calculate the square of 0.8. Remember that squaring means multiplying the base by itself: \(0.8 \times 0.8 = 0.64\).

Step 4: Apply the Negative Sign

Now that we have squared 0.8 to get \(0.64\), we multiply by -1 to account for the negative sign. Thus, \(-1 \times 0.64 = -0.64\).

Key Concepts

Negative Numbers

Negative numbers can be a bit tricky, especially when combined with operations like squaring. A negative number is simply a number smaller than zero, represented with a minus sign (-). Sometimes, this can confuse people when performing mathematical operations, like squaring.

• A negative number indicates the direction on the number line - to the left of zero.
• When a number has a negative sign in front of it, it impacts the result of operations performed on it.
• In expressions like \(-0.8^2\), the negative sign stays outside of the base for the exponent calculation resulting in a negative product after squaring the positive base.

When squaring a negative number like \(-0.8\), it's important to know if the negative is part of the base or not, which depends on the presence of parentheses. In \(-0.8^2\), the exponent only applies to \(0.8\), so we can think of this as \(-1 \times (0.8^2)\). Differentiating this helps prevent mistakes in calculations. Understanding these subtleties will greatly improve your handling of negative numbers in math.

Exponents

Exponents are a fundamental part of mathematics that indicate repeated multiplication of a number by itself. The number being multiplied is called the base, and the exponent tells you how many times to multiply the base.

• In \(a^n\), the number \(a\) is the base, and \(n\) is the exponent.
• Exponents are written as small numbers to the top right of the base.
• For example, \(0.8^2\) means \(0.8\) is used in the multiplication twice: \(0.8 \times 0.8\).

When you square a number, such as \(0.8\), you multiply it by itself, resulting in \(0.8 \times 0.8 = 0.64\). Understanding exponents is crucial as they are used in a wide range of mathematical contexts from simple calculations to complex algebraic expressions. Practicing the interpretation and operations involving exponents will bolster your skills in tackling such problems efficiently.

Order of Operations

The order of operations is a set of rules to determine the sequence in which different operations are carried out in a mathematical expression. This ensures consistency and correctness across different problems and solutions. Remembering the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)) can be helpful.

• Parentheses come first; solve anything inside them before moving on.
• Exponents come next, such as powers or roots.
• Multiplication and Division are performed after exponents, proceeding left to right.
• Addition and Subtraction are last, also from left to right.

In the expression \(-0.8^2\), even though it looks simple, applying the order of operations is crucial. After recognizing that \(-\) does not involve parentheses with \(0.8\), execute the exponent operation for \(0.8\), resulting in \(0.64\). Finally, apply the multiplication with \(-1\) as the last operation, yielding \(-0.64\). Understanding the order of operations ensures accuracy and helps avoid common pitfalls in mathematics.