Problem 19 - Algebra 1

Question

Write the verbal phrase as an algebraic expression. Use \(x\) for the variable in your expression. Two cubed divided by a number

Step-by-Step Solution

Answer

\((2^3) / x\)

Step 1: Understand the Verbal Phrase

First, understand what each part of the verbal phrase means. 'Two cubed' means two raised to the power of three. This is written as \(2^3\) in an algebraic expression. 'Divided by' tells us that we need to divide by something. Finally, 'a number' is asking for an unknown variable, which the exercise suggests to be \(x\).

Step 2: Translate into an Algebraic Expression

Using the understanding from Step 1, translate the verbal phrase into algebraic form. The result is \((2^3) / x\).

Key Concepts

Verbal Phrase Translation

Translating a verbal phrase into an algebraic expression is an essential skill in algebra that allows you to convert words into mathematical symbols. It's like turning a sentence into a math problem! Understanding the components of the phrase is crucial.
For example, consider the phrase "two cubed divided by a number." Let's break it down:

• "Two cubed" refers to the number 2, raised to the power of 3.
• "Divided by" means you'll be dividing one quantity by another.
• "A number" suggests an unspecified variable, often represented as \(x\) in algebra.

The goal is to translate these words into symbols, making it easier to solve or manipulate algebraically. In this case, "two cubed divided by a number" becomes \( (2^3) / x \).

Exponents

Exponents are a fundamental component of algebra and are used to denote repeated multiplication of a number by itself. When we say "two cubed," what do we mean?
This phrase tells you to multiply the number 2 by itself three times. Mathematically, this is expressed as:\[2 \times 2 \times 2\]When we simplify, 2 multiplied by itself is 4, and then multiplying by 2 again gives 8:\[2^3 = 8\]Exponents allow for a concise representation of larger numbers and simplify expressions. Whether you're dealing with perfect squares or very large powers, understanding exponents is key to mastering algebra.

Division in Algebra

In algebra, division is used to express the operation of dividing one number or expression by another. Within the verbal phrase "two cubed divided by a number," division plays a central role.
To divide algebraically, you place the dividend over the divisor just like you would when doing arithmetic with numbers:\[\frac{2^3}{x}\]Here, \(2^3\) is the dividend, and \(x\), the variable, is the divisor. This expression represents "8 divided by \(x\)," making it versatile and adaptable to various algebraic problems.
Understanding division in algebra helps to simplify expressions, solve for unknowns, and tackle complex equations by revealing relationships between different quantities.