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Chapter 1: Problem 68

Evaluate each expression for \(a=-1, b=3,\) and \(c=-4\) $$b^{2}-2 b-3$$

Short Answer

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Step by step solution

01

Substitute the values

Substitute the given values into the expression. For this expression, substitute -\(b = 3\)The expression becomes \(3^{2} - 2 \times 3 - 3\)
02

Evaluate the exponent

Calculate the square of 3. \(3^{2} = 9\)So the expression is now \(9 - 2 \times 3 - 3\)
03

Perform multiplication

Multiply 2 by 3. \(2 \times 3 = 6\)So the expression is now \(9 - 6 - 3\)
04

Perform subtraction

Subtract 6 from 9, and then subtract 3. \(9 - 6 = 3\), and \(3 - 3 = 0\)

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Key Concepts

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

Substitution in Algebra
One of the first steps to evaluating algebraic expressions is substitution. This means replacing variables in the expression with given numerical values.
In our exercise, we substitute the values provided for the variables. For example, for the expression \(b^{2} - 2b - 3\), you replace b with 3, because you were given that \(b = 3\).
Through this substitution, the expression becomes \(3^{2} - 2 \times 3 - 3\).
This step is critical because it turns a complex expression into a simpler arithmetic problem that is easier to manage.
  • Always make sure to substitute the values correctly and check your work to avoid mistakes.
Working with Exponents
After substitution, the next concept to tackle is exponents. An exponent tells you how many times to multiply a number by itself.
In our example, we have \(3^{2}\), which means multiplying 3 by itself once: \(3 \times 3 = 9\).
Exponents help to simplify expressions and reduce the number of multiplications you need to perform. When evaluating expressions, always handle exponents before moving on to multiplication and addition.
This is part of a broader set of rules called the order of operations, which ensures that mathematical expressions are simplified in a consistent and correct manner.
Order of Operations
To correctly evaluate expressions, we follow a specific set of rules called the order of operations. This helps to ensure that everyone solves the problem the same way and arrives at the same answer.
The basic order of operations can be remembered using PEMDAS:
  • P: Parentheses first
  • E: Exponents (powers and square roots, etc.)
  • M: Multiplication and D: Division (left-to-right)
  • A: Addition and S: Subtraction (left-to-right)
In the expression \(3^{2} - 2 \times 3 - 3\), we first evaluate the exponent to get 9. Then we perform the multiplication \(2 \times 3 = 6\), and finally, we handle the subtraction: \(9 - 6 - 3 = 0\).
Following the correct order prevents errors and ensures the right result.

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Most popular questions from this chapter

Solve each problem. Area of a lot. The algebraic expression for the area of a trapezoid, \(0.5 h\left(b_{1}+b_{2}\right),\) gives the area of the property shown in the figure. Find the area if \(h=150\) feet, \(b_{1}=260\) feet, and \(b_{2}=220\) feet.

Write an algebraic expression for each problem. Parthenon. To obtain a pleasing rectangular shape, the ancient Greeks constructed buildings with a length that was about \(\frac{1}{6}\) longer than the width. If the width of the Parthenon is \(x\) meters and its length is \(x+\frac{1}{6} x\) meters, then what is the perimeter? What is the area?

Simplify each expression. $$ 6-(x-4) $$

Simplify. $$ \frac{7 x^{3}-5 x^{3}-4}{-2} $$

Find each product or quotient. $$ \frac{-8 x^{2}}{4} $$
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