Problem 5 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the expression. $$ -4^{3} $$

Step-by-Step Solution

Answer

The expression \(-4^3\) is equal to \(-1 \times 4^3\), where \(4^3 = 4 \times 4 \times 4 = 64\). Therefore, the evaluated expression is \(-1 \times 64 = -64\).

Step 1: Identify Exponentiation Order

First, we need to identify the exponentiation order, which refers to the sequence in which we should compute the powers. In the given expression, the exponent is 3, and there is a negative sign in front of the base 4. The expression can be written as \(-1\times4^3\).

Step 2: Compute the Exponential Value

Now, compute the exponential value \(4^3\). Recall that an exponent indicates the base is to be multiplied by itself a number of times equal to the exponent. In this case, the base is 4, and the exponent is 3: \[4^3=4\times4\times4\]

Step 3: Simplify the Exponential Value

Next, simplify the exponential value by performing the multiplication: \[4\times4\times4=64\]

Step 4: Apply the Negative Sign

Finally, apply the negative sign to the simplified exponential value: \[-1\times64=-64\]

Step 5: Write the Final Answer

The evaluated expression is: \[-4^3=-1\times4^3=-1\times64=-64\]

Key Concepts

Exponential Expressions

Understanding exponential expressions is essential, as they are frequently used in math to represent growth, decay, and other phenomena that involve repeated multiplication. An exponential expression consists of two components: the base and the exponent. The base is the number being multiplied, and the exponent tells us how many times the base is used as a factor in the multiplication. For example, in the expression \(4^3\), 4 is the base and 3 is the exponent, indicating that 4 is multiplied by itself twice, as the exponent decrees one less multiplication than its actual value (i.e., \(4 \times 4 \times 4\)).

When dealing with exponential expressions, it's vital to remember that the multiplication of the base is done first before applying any other arithmetic operations such as addition, subtraction, or applying a negative sign. This is a key part of understanding the correct order of operations.

Negative Exponents

Negative exponents often cause confusion, but they follow a simple rule: a negative exponent means we take the reciprocal of the base and then apply the positive exponent. For instance, \(4^{-3}\) is the same as \(\frac{1}{4^3}\). This means that instead of multiplying 4 by itself 3 times, we would divide 1 by 4 multiplied by itself 3 times. A helpful way to remember this is that the negative sign in the exponent acts like an instruction to 'flip' the base into its reciprocal.

Dealing with Negative Signs Before Exponents

When a negative sign is present before an exponent, it affects the result. For example, \-4^{3}\ signifies that after calculating \(4^3\), we will apply the negative sign. The negative is not 'raised to the power', it is only applied to the result of the exponentiation of the base. Keep in mind that if the negative were inside the parentheses, like \(\left( -4 \right)^3\), it would mean the negative is part of the base to be raised to the third power.

Order of Operations

The order of operations is a fundamental concept to correctly simplifying expressions. Following the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), we can determine which operations to perform first. Exponents come right after calculations inside parentheses, before multiplication and division. This rule helps prevent misunderstandings in complex expressions and ensures consistency in solving math problems.

When we're given an expression such as \(\-4^3\), we first look for parentheses, then exponents. Since there are no parentheses, we move directly to the exponent. After calculating the power, we then apply the negative sign, because the multiplication (or applying the negative sign in this case) comes after evaluating the exponent in the order of operations. This sequence helps us reach the correct answer methodically and avoid common mistakes in calculation.