Problem 39 - Algebra for College Students

Question

Simplify each of the numerical expressions. $$(-3)^{2}-3(-2)(5)+4^{2}$$

Step-by-Step Solution

Answer

The simplified expression equals -5.

Step 1: Simplify the First Term

The expression begins with \((-3)^2\). This means to multiply \(-3\) by itself: \[-3 \times -3 = 9\].

Step 2: Simplify the Second Term

Next, we have \(-3(-2)(5)\). Start by multiplying the first two numbers:\(-3 \times -2 = 6\). Then multiply 6 by 5:\(6 \times 5 = 30\).

Step 3: Simplify the Third Term

Finally, solve \(4^2\) by multiplying 4 by itself:\(4 \times 4 = 16\).

Step 4: Combine All Terms

Now, substitute the simplified terms back into the expression:\(9 - 30 + 16\). Begin by performing the subtraction:\(9 - 30 = -21\). Then add 16:\(-21 + 16 = -5\).

Key Concepts

Exponents

Exponents express how many times a number, known as the base, is multiplied by itself. In essence, an exponent is shorthand for repeated multiplication. For instance, in the expression \((-3)^2\), the base is \(-3\), and the exponent is 2.

• This means you multiply \(-3\) by itself—which results in \(-3 \times -3 = 9\).
• Bear in mind that squaring a negative number results in a positive outcome.

Further, consider \(4^2\) from the original expression. Using the same logic, we multiply 4 by itself:

• Thus, \(4 \times 4 = 16\).

Remember, understanding exponents is crucial. It simplifies complex arithmetic expressions and reveals patterns in numbers. Always apply the base to itself as many times as the exponent indicates.

Order of Operations

Order of operations is a standard rule to solve mathematical expressions correctly. Without it, we could get different results for the same expression. Use the acronym PEMDAS to remember these steps:

• P: Parentheses
• E: Exponents
• M: Multiplication
• D: Division
• A: Addition
• S: Subtraction

For the expression \((-3)^{2}-3(-2)(5)+4^{2}\), we first tackled the exponents:

• \((-3)^2 = 9\)
• \(4^2 = 16\)

Then, we solve the multiplication:

• \(-3(-2)(5) = 30\)

Finally, we performed addition and subtraction in sequence:

• Evaluating the expression from left to right gives us \(9 - 30 + 16 = -5\).

Following PEMDAS ensures that we consistently arrive at the correct answer, avoiding common errors in complex arithmetic.

Multiplication

Multiplication is the mathematical operation of scaling one number by another. It’s an essential concept in math and appears frequently in problem-solving. When considering multiplication:

• It can be viewed as repeated addition.
• For instance, multiplying 2 by 3, represented as \(2 \times 3\), is the same as adding 2 three times \(2 + 2 + 2 = 6\).

In our exercise, we encountered a slightly complex multiplication situation:

• \(-3(-2)(5)\): Here, start by multiplying two terms first, then use the product to multiply with the third.
  • \(-3 \times -2 = 6\) and then \(6 \times 5 = 30\).
• Notice that multiplying two negative numbers yields a positive number. So be attentive to the signs of the numbers involved.

By understanding multiplication's basics and nuances, tackling complex numerical expressions becomes more manageable.