Problem 8 - Elementary Algebra

Question

For Problems \(1-20\), find the value of each numerical expression. For example, \(2^{4}=2 \cdot 2 \cdot 2 \cdot 2=16\). $$ -3^{4} $$

Step-by-Step Solution

Answer

The value of \(-3^4\) is \(-81\).

Step 1: Evaluate the Exponent

First, calculate the value of the expression with the exponent part. Since the expression is \(-3^4\), we need to raise \(-3\) to the power of 4. Calculate \((-3) \times (-3) \times (-3) \times (-3)\). Since we didn't enclose \(-3\) in parentheses in the equation \(-3^4\), it's actually computed as \(- (3 \times 3 \times 3 \times 3)\).

Step 2: Calculate 3 to the Power of 4

Calculate the value of \(3^4\).The calculation is as follows: \(3 \times 3 = 9\)\(9 \times 3 = 27\)\(27 \times 3 = 81\). So, \(3^4 = 81\).

Step 3: Apply the Negative Sign

Now apply the negative sign from the original expression before the exponent to the result from the previous step. The expression becomes \(-81\). Therefore, \(-3^4 = -81\).

Key Concepts

Negative Numbers

Negative numbers can often seem confusing, especially when combined with exponents. A negative number is any number less than zero. It is indicated by a minus sign (-) in front of the numeral. Understanding how these numbers work can make solving mathematical problems easier. When a negative number is involved with exponents, it is important to pay close attention to parentheses:

• If a negative number is raised to an exponent with parentheses, such as \((-3)^4\), it means that \(-3\) is multiplied by itself four times: \((-3) \times (-3) \times (-3) \times (-3)\), which results in a positive 81 because multiplying negatives in an even number of times results in a positive.
• Without parentheses, as in \(-3^4\), the negative sign is not part of the base being exponentiated. Instead, it indicates the opposite of the result of \(3^4\), leading to \(-81\).

Order of Operations

Order of operations is a fundamental principle that tells us which mathematical operations to perform first in a given expression. To simplify complex expressions accurately, one should follow the order of operations, often remembered by the acronym PEMDAS:

• P: Parentheses first
• E: Exponents (i.e., powers and roots)
• M/D: Multiplication and Division (from left to right)
• A/S: Addition and Subtraction (from left to right)

In the problem \(-3^4\), the expression without parentheses implies that only the \(3\) is raised to the power of \(4\), and the negative sign is applied afterward. So, one must calculate \(3^4\) first — following the Exponents rule — and then apply the negative sign to the result.

Numerical Expressions

Numerical expressions involve numbers and operation symbols but do not include an equality sign like equations do. Evaluating numerical expressions correctly ensures one arrives at the expected answer.In our example, the expression is \(-3^4\). Here, it's crucial to distinguish when parentheses are necessary to convey the intended meaning. Parentheses change how the expression is calculated:

• Without parentheses, the exponent only applies to \(3\), rendering \(3^4 = 81\), which then gives \(-81\) when the negative is applied post-evaluation.
• Whereas, with parentheses like in \((-3)^4\), the entire \(-3\) is raised to the power \(4\), computed as 81.

Thus, interpreting numerical expressions correctly requires understanding the hierarchy of operations and the impact of negative numbers on results.