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Chapter 1: Problem 88

Evaluate each expression. $$ -4(-3)^{2}+3(-3)-1 $$

Short Answer

Expert verified
The value of the expression is -46.

Step by step solution

01

Evaluate the Exponentiation

Start by evaluating the exponent in the expression \[(-3)^{2}\] which means multiplying \[-3\] by itself: \[-3 imes -3 = 9\]. The expression then becomes \[-4 imes 9 + 3 imes (-3) - 1\].
02

Multiply the Coefficients with the Terms

Next, multiply each coefficient by its respective term. Compute \[-4 imes 9 = -36\] and \[3 imes (-3) = -9\]. So, the expression simplifies to \[-36 - 9 - 1\].
03

Add and Subtract the Terms

Finally, add and subtract the resulting values left to right. Start by calculating the sum of \[-36 - 9 = -45\]\, then subtract \[-1\] \ from \[-45\]: \[-45 - 1 = -46\].

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Key Concepts

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

Negative Numbers
Negative numbers are numbers that are less than zero. They often represent values like debt or temperature below freezing. When working with negative numbers, especially in algebra, it's essential to pay attention to their impact on computations.
  • When multiplying two negative numbers, the result is positive.
  • When multiplying a negative number by a positive number, the result is negative.
  • Negative numbers can change the direction of adding or subtracting, which can be visualized on a number line.
In the expression \(-4(-3)^{2}+3(-3)-1\), it is important to distinguish between the negative sign that acts as a subtraction operation and the one that is part of the number itself.
Order of Operations
Order of operations is a set of rules that dictate the sequence in which calculations are performed. This is crucial in ensuring accurate results. The common acronym used to remember this order is PEMDAS:
  • Parentheses
  • Exponents
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)
In our exercise, the exponent \((-3)^{2}\) is evaluated before any multiplication or addition. Following this sequence accurately is vital, as getting the order wrong can lead to different answers. For example, computing multiplication or addition before tackling exponents is a common mistake. Always double-check that each component is addressed in the right order.
Multiplying Integers
Multiplying integers involves calculating the product of two whole numbers, which can be positive or negative. Here are some vital points:
  • Two positive integers multiply to give a positive product.
  • Two negative integers multiply to give a positive product.
  • A positive integer multiplied by a negative integer results in a negative product.
  • A negative integer multiplied by a positive integer again results in a negative product.
This is exactly what was done when simplifying the step \(-4 \times 9 + 3 \times (-3) - 1\). Understanding how to multiply integers properly is fundamental for handling more complex expressions successfully. Being confident in these basic rules will ensure accuracy in many algebraic operations.

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