Problem 51 - Fundamentals of Mathematics

Question

Determine the value of each of the powers. Use a calculator to check each result. \(8^{4}\)

Step-by-Step Solution

Answer

The value of \(8^4\) is 4096.

Step 1: Understand the Exponent

The expression given is \(8^4\). The number \(8\) is the base, and \(4\) is the exponent. An exponent of \(4\) means you need to multiply the base by itself four times.

Step 2: Write Out the Multiplication

Since \(8^4\) means \(8\) multiplied by itself 4 times, write out the multiplication: \(8 \times 8 \times 8 \times 8\).

Step 3: Calculate Step by Step

First, calculate \(8 \times 8 = 64\). Then, multiply the result by \(8\) again: \(64 \times 8 = 512\). Lastly, multiply \(512 \times 8 = 4096\).

Step 4: Verify With a Calculator

Use a calculator to compute \(8^4\): Input \(8\) and then the exponent function (usually marked as \(^\) or \(x^y\)) followed by \(4\). The calculator should display \(4096\), confirming your manual calculation.

Key Concepts

Base and Exponent

In mathematics, the terms "base" and "exponent" are integral to the concept of exponents. When you see an expression like \(8^4\), the number \(8\) is the base, which is the number being multiplied. The \(4\) is the exponent, indicating how many times you multiply the base by itself.

Understanding how bases and exponents work is crucial for evaluating powers correctly:

• The base tells you the number that is being used repeatedly. In our case, it's \(8\).
• The exponent gives you the number of times the base is used as a factor. Here, it's \(4\), meaning we multiply \(8\) by itself four times.

By knowing which number is the base and which is the exponent, you can confidently tackle powers in any mathematical problem.

Multiplication

Multiplication is at the heart of solving expressions involving exponents. The multiplication indicated by an exponent is a series of repetitive multiplications or repeated addition, which simplifies calculations when dealing with larger numbers.

For the expression \(8^4\), you write out the multiplication process as:

• First, multiply the base by itself: \(8 \times 8 = 64\).
• Next, take the product and multiply by the base again: \(64 \times 8 = 512\).
• Finally, multiply the result once more by the base: \(512 \times 8 = 4096\).

This step-by-step approach ensures accuracy and helps in understanding the process behind exponentiation.

Calculator Verification

Verifying your calculations using a calculator is an excellent way to ensure accuracy when dealing with exponents. With advancements in technology, most calculators are equipped to handle exponentiation, often with a dedicated exponent function button.

To verify \(8^4\) with a calculator, you can follow these steps:

• Turn on the calculator and input the base value, which is \(8\).
• Locate the exponent function, commonly represented as \(^\), \(x^y\), or similar symbols.
• Press the button and enter the exponent value, \(4\).
• Hit the "equals" button to see the result.

The display should show \(4096\), confirming our manual calculation. Using a calculator is a useful tool for quickly checking calculations and building confidence in your math skills.