Question

EQUATIONS AND INEQUALITIES Match the verbal sentence with its mathematical representation. The fourth power of \(x\) is 16

Short answer

The mathematical representation of the verbal sentence 'The fourth power of \(x\) is 16' is \(x^4 = 16\).

Step-by-step solution

Understanding the Terminology

The problem is asking for the fourth power of a number, \(x\), to be 16. This means \(x\) needs to be raised to the power of 4 and the result should be 16.

Translating the words into a Mathematical Representation

Now that what is required is understood, it can be translated into a mathematical equation. 'The fourth power of \(x\)' can be written as \(x^4\) and 'is 16' can be written as \(= 16\). So putting it all together, the mathematical representation becomes \(x^4 = 16\).

Key concepts

Equations and Inequalities

When we talk about equations in mathematics, we refer to statements that show the equality of two expressions, with one or more variables involved. In the case of inequalities, however, we are dealing with expressions that use symbols to indicate that one side is greater than, less than, or not equal to the other side.

For students grappling with these concepts, it's crucial to understand the dynamics of what makes them solid foundations in algebra. An equation like the fourth power of some number, say, our variable 'x', equaling 16, paints a picture of perfect balance. What we do to one side of this equation, if we aim to solve or manipulate it, we must do to the other to maintain equality.

In contrast, when we address inequalities, we're considering a range of possible solutions. A simple analogy would be to think of a seesaw. With equations, the seesaw is perfectly balanced, no matter how much weight you put, as long as it's equal on both sides. With inequalities, the seesaw tips to one side, symbolizing that one expression outweighs the other.

Mathematical Representation

Putting complex ideas into a language of numbers and symbols is what we mean by mathematical representation. It's a tool that allows students and mathematicians to convert verbal statements, such as 'the fourth power of x is 16,' into an equation that can be analyzed and solved.

Essentially, we are translating words into a universal language that adheres to specific rules and structures. This conversion process involves recognizing keywords and understanding how they relate to mathematical operations. For instance, the word 'power' refers to an exponential expression, and when we say 'is,' we introduce the concept of equality, which translates to the '=' sign in mathematical terms.

By mastering the art of translating verbal statements into equations, students lay the groundwork for further exploration in algebra, calculus, and beyond. It's important to remember that each part of a mathematical statement corresponds to a distinct part of the verbal instruction. The correct representation ensures that the integrity and intent of the original statement are preserved.

Exponential Expressions

Exponential expressions are another integral concept within algebra and mathematics as a whole. They involve numbers raised to a certain power, indicating repeated multiplication. Let's look at our original example, where we have the 'fourth power of x,' which is mathematically expressed as the exponential expression \(x^4\).

This indicates that 'x' is multiplied by itself four times (\(x\cdot x\cdot x\cdot x\)). It's a compact and efficient way of conveying potentially large numbers without writing out lengthy multiplication. Exponential expressions are critical when dealing with growth patterns, decay, compound interest, and in many other areas such as physics and engineering.

Understanding how to work with these expressions includes being able to recognize the base and the exponent, and knowing the laws that govern operations with exponents, such as the product rule, quotient rule, and power rule. Through practice and application, students can become proficient in managing these expressions and utilizing them to solve a wide array of problems.