Problem 31 - Prealgebra

Question

Write \(8 \cdot(k+3) \cdot(k+3)\) using exponents.

Step-by-Step Solution

Answer

The expression is written as \(8 \cdot (k+3)^2\).

Step 1: Understanding the Problem

We need to express the given expression, \(8 \cdot (k+3) \cdot (k+3)\), using exponents. Identifying the repeated terms is crucial for this process.

Step 2: Identifying the Repeated Factor

In the expression \((k+3) \cdot (k+3)\), the term \((k+3)\) is repeated twice. This suggests that we can use an exponent to simplify this part.

Step 3: Applying Exponents to the Expression

Since \((k+3)\) is repeated twice, it can be written as \((k+3)^2\). Now, let's rewrite the original expression.

Step 4: Rewriting the Entire Expression

Substitute \((k+3)^2\) back into the whole expression: \(8 \cdot (k+3)^2\). This incorporates the use of exponents to express the repeated multiplication.

Key Concepts

Expression Simplification

Simplifying expressions can make even the most complex math problems seem more approachable. It involves rewriting an expression in a more compact or understandable form, which often makes it easier to work with. When you look at an expression like \(8 \cdot (k+3) \cdot (k+3)\), it might seem complicated at first. The goal is to simplify this by finding patterns or repeated elements that can be expressed more succinctly.
The first step in simplifying expressions is to identify parts of the expression that are repeated or that can be combined. In this case, we spot that \((k+3)\) is being multiplied by itself. This repetition allows us to express the term using an exponent. Moreover, constants and variables are grouped or rearranged to streamline the expression, such as turning \(8 \cdot (k+3)^2\) into a more manageable form compared to writing out repeated terms.

• Identify repeating factors or terms.
• Use exponents to express repeated multiplication.
• Combine like terms or constants to further simplify.

Using simplification techniques helps reduce calculation errors and makes evaluating expressions quicker when working through equations or applying them in different mathematical contexts.

Repeated Multiplication

Repeated multiplication is the idea of multiplying a number or expression by itself several times. This is where exponents shine! If you see \((k+3) \cdot (k+3)\), it indicates you are multiplying \((k+3)\) by itself twice. This can be time-consuming to write out in full every time, especially if there are even more repetitions involved.
Exponents are used to simplify this kind of repeated multiplication. Instead of writing the same term multiple times, you write it once with an exponent that tells you how many times the multiplication should occur. For example:

• \((k+3) \cdot (k+3)\) can be rewritten as \((k+3)^2\)
• This shows the base \((k+3)\) is used in multiplication twice.

By using exponents, you reduce the complexity and length of expressions. This not only makes it easier to read but also simplifies performing any subsequent mathematical operations on the expression.

Mathematical Notation

Mathematical notation is the language of math that uses symbols and shorthand to represent numbers, operations, and concepts. It allows mathematicians to convey complex ideas efficiently and unambiguously. In the expression \(8 \cdot (k+3) \cdot (k+3)\), mathematical notation helps in simplifying this expression into \(8 \cdot (k+3)^2\), which employs exponents to show the repeated multiplication of the term \((k+3)\).
There are many benefits to using mathematical notation:

• Clarity: Notation can make an expression clearer and easier to understand at a glance.
• Precision: It provides exact information about operations and procedures.
• Efficiency: Provides a way to quickly convey complex operations without extensive text.

Adopting proper notation like exponents ensures that expressions remain straightforward, reducing the chance of mistakes and making calculations more straightforward as you progress with math learning. Understanding and using mathematical notation is essential for successfully navigating and solving more advanced mathematical concepts.