Problem 66 - Algebra A Combined Function

Question

Perform the indicated operation. \(-3^{4}\)

Step-by-Step Solution

Answer

The result of \(-3^{4}\) is \(-81\).

Step 1: Identify the Base and Exponent

In the expression \(-3^{4}\), the base is \(3\) and the exponent is \(4\). The negative sign is not included in the base here as it is outside the exponentiation.

Step 2: Calculate the Power

Compute the power by multiplying the base by itself the number of times indicated by the exponent. Therefore, calculate \(3^4 = 3 \times 3 \times 3 \times 3 = 81\).

Step 3: Apply the Negative Sign

Finally, apply the negative sign to the result obtained from the power calculation. The expression \(-3^4\) becomes \(-81\).

Key Concepts

Exponents and Powers

Exponents and powers are fundamental concepts in mathematics, often used to simplify repeated multiplication. An exponent indicates how many times a base number is multiplied by itself.
For instance, in the expression \(3^4\), we call \(3\) the base and \(4\) the exponent.
Here's what this looks like:

• \(3^4 = 3 \times 3 \times 3 \times 3 = 81\)

This means we multiply \(3\) by itself \(4\) times. Knowing how to handle exponents properly can make it much easier to handle more complex mathematical problems. Powers, another term for these expressions, indicate the result of the exponent operation.

Order of Operations

Order of operations is crucial in mathematics to ensure the correct calculation of expressions. It dictates the sequence in which operations should be performed.
When dealing with an expression like \(-3^4\), it is important to follow the order of operations:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

This is often remembered by the acronym PEMDAS. In the exercise \(-3^4\), the exponent operation must be performed before applying the negative sign. This is because exponents take precedence over addition and subtraction, including negative signs. By evaluating \(3^4\) first, you ensure an accurate result, which is then negated, resulting in \(-81\).

Negative Numbers

Negative numbers are numbers less than zero and are represented by a negative sign \((-\)). Understanding how to work with negative numbers is essential in many areas of math.
In the expression \(-3^4\), the negative sign operates independently of the exponent multiplication.
This means you first calculate the exponent part \(3^4 = 81\), and then apply the negative sign, making it \(-81\).
Remember:

• A negative sign before a number changes the sign of the product when multiplied.
  
• Without parentheses, \(-a^b\) implies \(-(a^b)\), not \((-a)^b\).
  

Getting comfortable with these nuances helps in calculating accurate results, especially when multiple operations are involved.