Problem 39 - Fundamentals of Mathematics

Question

Determine the value of each of the powers. Use a calculator to check each result. \(3^{4}\)

Step-by-Step Solution

Answer

The value 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. This means we need to multiply the base, 3, by itself 4 times.

Step 2: Express the Power as Repeated Multiplication

Write out the expression \(3^4\) as a repeated multiplication: \(3 \times 3 \times 3 \times 3\).

Step 3: Calculate the Repeated Multiplication

Perform the multiplication: \(3 \times 3 = 9\), then \(9 \times 3 = 27\), and finally \(27 \times 3 = 81\). So, \(3^4 = 81\).

Step 4: Verify Using a Calculator

Use a calculator to enter \(3^4\). You should input '3', then the exponentiation function often labeled as '^', followed by '4'. Ensure the result matches 81.

Key Concepts

Base and Exponent

Exponentiation is a mathematical operation that involves two main components: the base and the exponent. In the expression \(3^4\), the base is 3, which is the number that will be multiplied by itself multiple times. The exponent is 4, which tells us how many times the base is used as a factor in the multiplication. This concept is crucial because the exponent determines the number of times the base is repeated. So, whenever you see an expression like this, it's telling you to take the base number and multiply it by itself, as many times as the exponent indicates. This not only shows the number of factors but also signifies the power to which the base is raised. Understanding the roles of the base and exponent simplifies the process of working with powers and exponents in mathematics.

Repeated Multiplication

When you have an expression like \(3^4\), it can be broken down into repeated multiplication. This means we take the base, which is 3, and multiply it by itself as many times as the exponent tells us—four times in this case. Therefore, \(3^4\) is equivalent to \(3 \times 3 \times 3 \times 3\). This form is more explicit and makes it easier to perform multiplication step-by-step:

• First, multiply the first two 3s: \(3 \times 3 = 9\).
• Then, multiply the result by the next 3: \(9 \times 3 = 27\).
• Finally, multiply again by the last 3: \(27 \times 3 = 81\).

Therefore, \(3^4 = 81\). This method shows how exponents can save us from writing long multiplication chains and helps streamline calculations.

Calculator Verification

After performing the multiplication manually, it's always a good idea to verify your results using a calculator. Most calculators have a button for exponentiation, often labeled as '^' or 'EXP'. To confirm \(3^4 = 81\), enter the following sequence:

• Input the base number '3'.
• Press the exponentiation key, which might look like '^'.
• Enter the exponent '4'.
• Finally, press 'equals' or 'enter'.

If done correctly, the calculator should display 81, verifying our manual calculation. This step is essential as it confirms accuracy, especially when dealing with larger numbers or higher exponents where manual multiplication can be prone to error. It's a practical way to couple technology with math skills, ensuring correct results every time.