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Chapter 2: Problem 71

Evaluate the expression. $$79-v^{3} \text { when } v=4$$

Short Answer

Expert verified
The evaluated expression is 15.

Step by step solution

01

Substitute the given value into the expression

Replace \(v\) in the expression \(79 - v^3\) with the given value \(4\). That gives \(79 - 4^3\).
02

Simplify the expression

Calculate the cube of 4 which is \(4^3 = 64\). Substitute this value into expression, resulting in \(79 - 64\).
03

Subtract to find the final result

Now subtract 64 from 79, which equals 15. So, \(79 - 4^3 = 15\).

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Key Concepts

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

Substitution
Substitution is the process of replacing a variable with a given number. This technique simplifies an algebraic expression and allows you to calculate its value. In the exercise, we were given the expression \(79 - v^3\), where \(v = 4\). To evaluate the expression, we substitute:
  • Replace \(v\) with \(4\) in the expression.
  • This results in \(79 - 4^3\).
It's a straightforward but crucial step that transforms an abstract expression into something concrete we can work with. Getting comfortable with substitution helps simplify and make sense of what might seem like complex algebraic expressions. It's much like swapping a placeholder with the actual number when you substitute.
Exponentiation
Exponentiation is a mathematical operation involving numbers, where one number, called the base, is raised to the power of another number, the exponent. In our exercise, we had the term \(v^3\). This is an example of exponentiation, where:
  • The base \(v\) is replaced by \(4\).
  • The 3 is the exponent, indicating that \(4\) is multiplied by itself twice more: \(4 \times 4 \times 4\).
  • This results in \(4^3 = 64\).
Exponentiation is a step that transforms expressions with powers into everyday arithmetic numbers, which can then be added or subtracted as needed. A good understanding of how to perform exponentiation is fundamental for solving many algebraic problems.
Arithmetic Operations
Arithmetic operations are the basic calculations we use every day: addition, subtraction, multiplication, and division. In the context of the given exercise, the arithmetic operation we perform is subtraction. Once we have substituted and simplified the expression from \(79 - v^3\) to \(79 - 64\), the next step is:
  • Perform the subtraction: \(79 - 64\).
  • This further simplifies the expression to \(15\), which is the final result.
Arithmetic operations are crucial as they allow you to combine and reduce layers of numbers and variables to find exact solutions. Proficiency in these operations enables you to handle more complex algebraic expressions with ease.

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Most popular questions from this chapter

COMBINING LIKE TERMS Apply the distributive property. Then simplify by combining like terms. $$ 7 x-3 x(x+1) $$

RECIPROCALS Find the reciprocal. $$ \frac{6}{11} $$

LOGICAL REASONING Decide whether the statement is true or false If false, rewrite the right-hand side of the equation so the statement is true. $$ \frac{2}{9}\left(\frac{1}{3}-\frac{4}{9}\right) \stackrel{?}{=} \frac{2}{9}\left(\frac{1}{3}\right)-\frac{2}{9}\left(\frac{4}{9}\right) $$

Use the distributive property and mental math to simplify the expression. $$ 3 x^{2}+4 x+8-7 x^{2} $$

SIMPLIFYING EXPRESSIONS Simplify the expression by combining like terms. $$ -12 y+5 y $$
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