Problem 39 - Introductory Algebra for College Students

Question

In Exercises \(29-72,\) use the order of operations to simplify each expression. $$3(-2)^{2}-4(-3)^{2}$$

Step-by-Step Solution

Answer

-24

Step 1: Calculate Exponents

The exponent operation indicates that the number should be multiplied by itself the given number of times. In this case, \(-2^{2}\) simplifies to 4 and \(-3^{2}\) simplifies to 9.

Step 2: Apply Multiplication Principle

Multiplication is done before subtraction in order of operations. Multiply 3 by 4 to get 12 and also multiply 4 by 9 to get 36.

Step 3: Perform Subtraction

Finally, subtract the result from step 2. Subtract 36 from 12 to get \(-24\).

Key Concepts

Understanding Exponents

Exponents are a key part of algebra and mathematics as a whole. An exponent tells you to raise a number to a certain power, meaning you multiply the number by itself as many times as indicated by the exponent. When you see an expression like \((-2)^{2}\), it means \((-2) \times (-2)\).

• The base number is -2.
• The exponent is 2.
• The result is 4 because a negative number multiplied by itself an even number of times results in a positive number.

This rule also applies to \((-3)^{2}\), which equals \((-3) \times (-3) = 9\).Exponents help in simplifying expressions by effectively condensing repeated multiplication into a shorthand notation. This is particularly useful when dealing with complex algebraic expressions.

The Role of Multiplication in Order of Operations

Multiplication is one of the critical operations when following the order of operations, often remembered using the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). After calculating any exponents in an expression, you should move on to multiplication or division, from left to right.For instance, given the expression \(3 \times 4 - 4 \times 9\), we first handle the multiplication:

• Multiply 3 by 4 to obtain 12.
• Multiply 4 by 9 to get 36.

This reaffirms the necessity to multiply before addressing addition or subtraction. In the context of our exercise, respecting this order ensures accurate simplification of algebraic expressions.

Decoding Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and operations. Simplifying them requires understanding and applying the correct order to calculate accurately and efficiently.In a typical algebraic expression like \(3(-2)^{2} - 4(-3)^{2}\), consider the following:

• Simplify exponents first, as already explained, simplifying to \(3 \times 4 - 4 \times 9\).
• Perform multiplication next, yielding 12 and 36.
• Complete the simplification with subtraction, \(12 - 36\), giving the final result of \(-24\).

Algebraic expressions become much manageable when the order of operations is diligently followed. By understanding these core components, mastery over more complex equations becomes achievable.