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Chapter 2: Problem 15

Perform the operations. Simplify the result when possible. $$3+4^{3}(-5)$$

Short Answer

Expert verified
The result of the operation \(3+4^{3}(-5)\) is -317.

Step by step solution

01

Evaluate the Exponent

Start the exercise by handling the exponent operation. This means calculating \(4^3\), which equals 64.
02

Multiplication

The next operation in the order of operations is multiplication. The problem requires multiplying 64 (the result of the previous step) with -5. This results in -320.
03

Addition

The final operation that needs to be performed is addition. The problem requires adding 3 to -320 (result of the multiplication step). This results in -317.

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Key Concepts

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

Understanding Exponents
Exponents are a mathematical tool used to express how many times a number, known as the base, is multiplied by itself. When dealing with exponents, it's important to interpret them correctly to simplify expressions accurately. In the expression \(4^3\), 4 is the base, and 3 is the exponent. This means that 4 should be multiplied by itself three times: \(4 \times 4 \times 4\).
  • In this case, \(4 \times 4 = 16\).
  • Then, \(16 \times 4 = 64\).
Thus, \(4^3 = 64\). Exponents are powerful because they allow us to handle large numbers quickly and efficiently. Remember that correctly calculating exponents is crucial for solving expressions accurately in more complex problems.
The Role of Multiplication in Order of Operations
Multiplication is an arithmetic operation that involves combining equal groups. In the context of the order of operations, multiplication must be completed after exponents, but before addition and subtraction, unless specified by parentheses. In the expression from the original exercise, once the exponent \(4^3\) is calculated as 64, the next step is multiplication: \(64 \times (-5)\).
  • The negative sign in front of the 5 indicates that this is a negative multiplication.
  • Therefore, \(64 \times (-5) = -320\).
Understanding how to handle positive and negative numbers during multiplication is key. A negative times a positive yields a negative result. Always follow the rules of the order of operations to ensure accuracy in resolving expressions.
Simplifying with Addition
Addition is often one of the final steps when simplifying an expression. It combines values to yield a single sum. In the problem we examined, after calculating the multiplication to get -320, the final step is to add 3 to -320.
  • Think of addition as moving on a number line. If you are at -320 and you want to add 3, you move three steps towards zero.
  • This results in: \(-320 + 3 = -317\).
When adding a positive number to a negative number, you are effectively reducing the absolute value of the negative number. Understanding how positive and negative values interact is essential for solving these types of problems correctly.

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