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Chapter 1: Problem 47

Perform each operation. SEE EXAMPLE \(2 .\) (OBJECTIVE 1 ) $$2(-1)^{2}-3(-2)^{2}$$

Short Answer

Expert verified
The simplified result of the expression is 14.

Step by step solution

01

Evaluation of exponents

Begin with evaluating the exponents: \((-1)^{2}\) becomes 1 and \((-2)^{2}\) becomes 4.
02

Perform multiplication

After the exponents are simplified, perform the multiplication. 2 times 1 becomes 2 and -3 times 4 becomes -12.
03

Subtract the two results

Finally, subtract -12 from 2 to get the final answer.

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Key Concepts

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

Exponents
Understanding exponents is crucial when dealing with algebraic expressions. An exponent, also known as a power, indicates how many times a number, called the base, is multiplied by itself. For example, in the expression \(a^n\), \(a\) is the base and \(n\) is the exponent. When the base is negative, the outcome depends on whether the exponent is even or odd. An even exponent results in a positive value, while an odd exponent results in a negative value. For instance, \( (-1)^2 \) equals 1 because \( -1 \) multiplied by itself once results in a positive 1. Similarly, \( (-2)^2 \) equals 4.

In summary, when dealing with exponents, remember to:
  • Identify the base and the exponent.
  • Apply the exponent by multiplying the base by itself the specified number of times.
  • Consider the sign of the base, especially for negative bases, as this will influence the sign of the result when the exponent is even or odd.
Multiplication in Algebra
When multiplying algebraic terms, certain rules help ensure accuracy. In algebra, multiplication is commutative, meaning the order of the terms doesn't affect the result. For example, \(3 \times 4\) is the same as \(4 \times 3\) and yields 12. When numbers are multiplied with variables, the numerical coefficients are multiplied together while the variable remains the same, such as \(2a \times 3a = 6a^2\).

Here are some key points to remember during multiplication in algebra:
  • Multiply coefficients (numerical parts) with each other.
  • Multiply variables by considering exponents (using the power rule for same bases).
  • Maintain the proper signs (+ or -) based on multiplication rules for positive and negative numbers.
Order of Operations
The order of operations is a fundamental concept in algebra that dictates the sequence in which different mathematical operations should be performed to accurately solve an expression. The common acronym for remembering this order is PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right).

To avoid confusion while performing operations, follow these rules:
  • Start with calculations inside parentheses or brackets.
  • Next, evaluate exponents or powers.
  • Proceed with multiplication and division as they appear from left to right.
  • Finally, conduct addition and subtraction as they appear from left to right.

In the given exercise, we first evaluate the exponents, followed by multiplication with their respective coefficients, and conclude with the subtraction of the products.

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Most popular questions from this chapter

Each TV a retailer buys costs \(\$ 3.25\) per day for warehouse storage. What does it cost to store 37 TVs for three weeks?

Which property of real numbers justifies each statement? $$9 \cdot \frac{1}{9}=1$$

A girl had \(d\) dollars, and her brother had \(\$ 5\) more than three times that amount. How much did the brother have?

Explain why there is no greatest natural number.

Which property of real numbers justifies each statement? $$x(y+z)=(y+z) x$$
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