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Chapter 2: Problem 18

In Exercises 11-18, identify the coefficient of the term. $$ \pi t^{4} $$

Short Answer

Expert verified
The coefficient of the term \( \pi t^{4} \) is \(\pi\).

Step by step solution

01

Identify the Coefficient

The term is \( \pi t^{4} \). The coefficient of a term is the numerical factor of the term. Here the coefficient is the number \(\pi\) which is not attached to any variable.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Terms

An algebraic term is an expression that includes numbers, variables (like x or y), and possibly exponents that indicate the powers to which variables are raised. Terms are the building blocks of algebraic expressions, like the words that make up a sentence. Just as sentences describe ideas, algebraic expressions describe numerical relationships.

When you look at an algebraic expression, you'll notice it's made up of several terms separated by '+' or '-' signs. For instance, in the expression 3x + 2y - 7, there are three terms: 3x, 2y, and -7. Each term tells part of the story of the expression.

Understanding Individual Terms

  • Variables: Letters that stand for numbers not yet known.
  • Coefficients: Numbers multiplied by the variables in a term.
  • Constant Terms: Numbers without any variables, which act as a standalone value in an expression.

Algebraic terms can become quite complex, but the fundamental idea is to understand that each term is made up of these key elements. They work together to give each term鈥攁nd thus, the entire expression鈥攊ts unique value and meaning.

Numerical Coefficient

A numerical coefficient is the number that multiplies a variable within an algebraic term. It quantitatively describes how many times the variable is included in the term. Most often, coefficients are placed directly in front of the variables they modify, with the assumption of multiplication between them.

For example, in the term 5x, the numerical coefficient is 5. This tells us that x is present five times. If a term like x appears without a visible coefficient, it has an implied numerical coefficient of 1, because variables alone effectively mean 'one of' themselves.

Coefficients in Different Contexts

  • In -3y, the coefficient is -3, implying three times the variable y with a negative sign.
  • The term 0.25z has a numerical coefficient of 0.25, indicating a fraction of the variable z is used.

Coefficients can also be irrational numbers, like \(\pi\), or even complex numbers in advanced mathematics. The key is to recognize that coefficients are the pieces that scale variables and, therefore, they play a crucial role in determining the overall value of an algebraic term.

Variable Exponentiation

Variable exponentiation is the process of raising a variable to a power, which is to say, using an exponent to denote repeated multiplication. The exponent tells us how many times to multiply the variable by itself. It's a concise way to represent a potentially long string of multiplications.

For instance, take the term x^3. The exponent here is 3, which means x is multiplied by itself two additional times: \(x \times x \times x\). This is a much more efficient way of expressing large-scale multiplications than writing out every individual factor.

Working with Exponents

  • Exponents only apply to the base they are with; for example, in 2x^3, the exponent applies just to x, not to 2.
  • When there are no exponents listed, it is assumed to be 1, as in y^1, which is simply y.
  • Variables can be raised to power just like numbers, and the rules of exponents apply to variables in the same manner.

Understanding how to work with variable exponentiation is essential for simplifying expressions, solving algebraic equations, and working with polynomials. It鈥檚 a fundamental concept in algebra that allows for the representation of growth patterns, area calculations, and much more within mathematics.

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Most popular questions from this chapter

In Exercises 33-38, justify each step of the equation. Then identify any properties of equality used to solve the equation. $$ \begin{aligned} 5 x+12 &=22 \\ 5 x+12-12 &=22-12 \\ 5 x &=10 \\ \frac{5 x}{5} &=\frac{10}{5} \\ x &=2 \end{aligned} $$

In Exercises 51-54, determine whether each value of \(x\) is a solution of the equation. \(\frac{5}{x-1}+\frac{1}{x}=5\) (a) \(x=3\) (b) \(x=\frac{1}{6}\)

In Exercises 17-26, determine whether an algebraic expression or an algebraic equation is given. $$ \frac{1}{2} x $$

In Exercises 33-38, justify each step of the equation. Then identify any properties of equality used to solve the equation. $$ \begin{aligned} x-8 &=3 \\ x-8+8 &=3+8 \\ x &=11 \end{aligned} $$

Determine whether order is important when translating each verbal phrase into an algebraic expression. Explain. (a) \(x\) increased by 10 (b) 10 decreased by \(x\) (c) The product of \(x\) and 10 (d) The quotient of \(x\) and 10
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