Problem 40 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the order of operations to simplify each expression. $$5(-3)^{2}-2(-4)^{2}$$

Step-by-Step Solution

Answer

The simplified version of the expression is 13.

Step 1: Solving the powers

Start by simplifying the powers. Therefore, \((-3)^{2}\) becomes 9 and \((-4)^{2}\) becomes 16.

Step 2: Multiplication and Subtraction

Now, multiplying 5 by the result of first parenthesis and 2 by the result of the second parenthesis, we obtain: \[5*9 - 2*16\] which simplifies to \[45 - 32\]

Step 3: Final Simplification

Finally, subtracting 32 from 45 to simplify fully, we get \[45-32 = 13\].

Key Concepts

Simplifying Expressions

Simplifying expressions involves breaking them down to their simplest form while maintaining their original value. In the context of our example, it involves a blend of mathematical operations like powers, multiplication, and subtraction. The idea is to follow a logical sequence established by the order of operations (PEMDAS/BODMAS).

• Parentheses/Brackets
• Exponents/Orders (i.e., powers and square roots, etc.)
• MD Multiplication and Division (from left to right)
• AS Addition and Subtraction (from left to right)

Understanding this sequence is essential for solving expressions accurately. In our example, we prioritize solving the exponent components first before addressing multiplication and finally handling subtraction. Taking these steps ensures the expression is broken down methodically for accurate simplification.

Powers and Exponents

Powers and exponents are fundamental in simplifying mathematical expressions, as they dictate the operation of multiplying a number by itself a specific number of times. When you see something like \((-3)^{2}\), this means multiply -3 by itself. Therefore:
\( (-3) \times (-3) = 9 \)
For \((-4)^{2}\), the same rule applies:
\( (-4) \times (-4) = 16 \)
Here, it's crucial to note that the negative sign inside the parentheses is also subjected to the power, influencing the final value. This process highlights the importance of handling exponents early in the problem-solving phase, according to the order of operations.

Multiplication and Subtraction

Once the powers are resolved, you move on to multiplication, as it appears next in the sequence of operations. Take the resolved powers and their coefficients from the expression to perform multiplication. From our exercise:

• Multiply 5 by the result of \((-3)^{2}\), which is \(5 \times 9 = 45\).
• Multiply 2 by the result of \((-4)^{2}\), which is \(2 \times 16 = 32\).

Next, handle subtraction. Using the results from multiplication:
\(45 - 32\)
Perform the subtraction operation from left to right to arrive at the simplified form of the expression.
Thus, \(13\) is the final result, showcasing how understanding each component's role helps accurately simplify complex expressions.