Question

Write the verbal sentence as an equation, or an inequality. A number \(x\) squared plus forty-four is equal to the number \(x\) to the fourth power times three.

Short answer

The mathematical equation that accurately represents the given verbal sentence is \(x^2 + 44 = 3x^4\).

Step-by-step solution

Understand the Problem

The first step is to understand the given problem. Make sure you understand what the verbal sentence is representing in terms of a mathematical idea.

Translate the Sentence into a Mathematical Equation

Begin with the first part of the sentence: 'A number x squared'. This translates to \(x^2\). The next part is 'plus forty-four,' which translates to '+ 44'. So far, with these two parts, the mathematical equation is \(x^2 + 44\). The sentence continues with 'is equal to,' which translates to '=' in mathematics. The last part of the sentence is 'the number x to the fourth power times three,' which translates to \(3x^4\). Combining all of these together, the mathematical equation derived from this sentence is \(x^2 + 44 = 3x^4\).

Check Your Equation

Double-check your equation to make sure all components of the verbal sentence have been represented accurately. Remember to check it makes sense in the context of the originally provided verbal sentence. If the equation seems accurate, then your translation has been successful.

Key concepts

Algebraic Expressions

When dealing with algebraic expressions, it's crucial to understand that they are collections of numbers, variables, and operations. These components, when combined, represent a value. An algebraic expression doesn't need to equal anything—it’s simply a phrase that can take on different values depending on the variable.

Let's consider the verbal description 'A number x squared plus forty-four'. 'A number squared' indicates that the variable, which we are calling 'x', is being multiplied by itself. The 'plus forty-four' indicates an addition of forty-four to this squared number. Together, they form the algebraic expression \(x^2 + 44\).

Equation Writing

Writing equations is a key skill in algebra that involves translating words into mathematical symbols. The basic components of an equation include the variable, coefficients, operations (such as addition and multiplication), and the equals sign. For the verbal sentence provided, 'is equal to' signifies the presence of an equals sign, creating a balance between two expressions.

By recognizing that 'the number x to the fourth power times three' refers to the product of three and the fourth power of x, or \(3x^4\), you can set up an equation from the verbal statement: \(x^2 + 44 = 3x^4\). This constitutes a standard form for an algebraic equation with equality between two expressions.

Mathematical Translation

Mathematical translation involves converting words into a language that mathematics understands—equations and expressions. To translate effectively, one must identify keywords and phrases that correspond to mathematical operations and relationships.

• 'A number x squared' is understood as \(x^2\).
• 'Plus' indicates addition.
• 'Is equal to' means we are dealing with an equation, represented by '='.
• 'The number x to the fourth power times three' can be translated to \(3x^4\).

Stringing these together, as shown in the step-by-step solution, leads to an accurate portrayal of the verbal sentence through the lens of mathematical symbolism.

Exponents and Powers

When verbal sentences mention terms like 'squared' or 'to the fourth power', these refer to exponents and powers. An exponent tells you how many times to use the base (the number being multiplied) as a factor in a multiplication. For example, when 'x squared' is mentioned, it translates to \(x^2\), implying that x is being multiplied by itself once: \(x * x\).

The same goes for 'to the fourth power'. It’s written as \(x^4\) and means that x is a factor four times: \(x * x * x * x\). Multiplying it by three, as the verbal sentence specifies, gives us \(3 * x^4\) or, more simply, \(3x^4\). Understanding how to work with exponents and powers is fundamental when translating such phrases into algebraic equations.