Problem 1 - Beginning and Intermediate Algebra

Question

The expression \(8 x^{2}\) means A. \(8 \cdot x \cdot 2\) B. \(8 \cdot x \cdot x\) C. \(8+x^{2}\) D. \(8 x \cdot 8 x\)

Step-by-Step Solution

Answer

B. \(8 \times x \times x\)

Step 1: Understand the expression

The expression given is \(8 x^{2}\). This notation means 8 times the square of \(x\).

Step 2: Expand the expression

Square of \(x\) is denoted as \(x^{2}\), which means \(x \times x\). So, the expression \(8 x^{2}\) means 8 times \(x \times x\).

Step 3: Identify the correct option

Based on the expanded expression, \(8 x^{2}\) means \(8 \times x \times x\). Now match this with the given options:A. \(8 \times x \times 2\) (Incorrect)B. \(8 \times x \times x\) (Correct)C. \(8 + x^{2}\) (Incorrect)D. \(8 x \times 8 x\) (Incorrect)Therefore, the correct option is B.

Key Concepts

algebraic notation

Understanding algebraic notation is crucial for simplifying and solving algebraic expressions.
It uses symbols and letters to represent numbers and operations. In the expression given, \(8x^{2}\), let's break it down:

• The number 8 is a coefficient.
• The letter \(x\) is a variable.
• The superscript 2 (in \(x^{2}\)) is an exponent.

Effectively, it tells us that 8 is multiplied by the square of \(x\). Recognizing and correctly interpreting these notations is the first step to solving algebra problems.

exponentiation

Exponentiation is a mathematical operation involving two numbers. Here, the base is \(x\), and the exponent is 2 in \(x^{2}\).
This means we multiply the base by itself, which gives us \(x \times x\) when the exponent is 2. Exponents can be larger or smaller and understanding them helps simplify expressions.
Some rules to remember:

• \(x^{n} = x \times x \times \ldots \times x\), n times
• \(1^{n} = 1\), any number to the zero power is 1
• \(x^{m} \times x^{n} = x^{m+n}\)
• \((x^{m}){^{n}} = x^{m \times n}\)

Here, \(8x^{2}\) means \(8 \times x \times x\).

multiplication rules

Multiplication rules help to simplify expressions appropriately. For algebra:

• Combine like terms.
• Use the distributive property \(a(b + c) = ab + ac\).
• Multiply coefficients separately from variables.

In the problem, \(8x^{2}\) means 8 multiplied by \(x^{2}\) or \(x \times x\).
Applying multiplication rules, focus first on the coefficient (8) and then exponent multiplication resulting in \(8 \times x \times x\).

terms in expressions

An expression is made up of different terms. Terms are separated by plus (+) or minus (−) signs. In the given expression, \(8x^{2}\) is a single term.
Each term can include:

• Coefficients (numbers like 8)
• Variables (letters like x)
• Exponents (like 2 in \(x^{2}\))

Recognizing and breaking down terms helps in manipulating and solving algebraic expressions. Here, \(8x^{2}\) tells us that 8 is the coefficient, \(x\) is the variable, and the entire expression represents 8 times the square of \(x\). This identification is critical for correctly simplifying and solving algebra problems.