Problem 51 - Algebra 1

Question

Use a calculator to evaluate the power. For keystroke help see Student Help box on page 11. $$ 8^{6} $$

Step-by-Step Solution

Answer

The result of the operation \(8^{6}\) is 262144.

Step 1: Understand the Power Operation

The expression \(8^{6}\) refers to the number 8 multiplied by itself 6 times. In other words, \(8^{6} = 8 × 8 × 8 × 8 × 8 × 8\).

Step 2: Use a Calculator for the Power Operation

On the calculator, type '8', then press the 'exponent' button (this might be '^' on some calculators), followed by '6'. Lastly, press the 'equals' button. Your calculator will perform the operation and display the result.

Step 3: Record the result

Write down the number displayed on your calculator. This number is the result of the operation \(8^{6}\).

Key Concepts

Powers and Exponents

Understanding powers and exponents is fundamental in mathematics. Exponents are a concise way to express repeated multiplication of the same number. When we write an expression like 8^6, it's an exponentiation indicating that the number 8, known as the base, is multiplied by itself 6 times, the exponent. The exponent dictates the number of times the base appears in the multiplication.

Let's break it down:

• The base is the number we are going to multiply repeatedly.
• The exponent, located in the superscript position, tells us how many times to multiply the base by itself.
• The result of exponentiation is known as a 'power.' So, 8^6 is read as 'eight to the sixth power.'

Properly understanding the concept of raising a number to a power is crucial, as it simplifies complex calculations and serves as the foundation for more advanced topics such as scientific notation, polynomial expressions, and exponential growth.

Calculator Usage for Exponents

Calculators are invaluable tools for simplifying complex mathematical operations, especially when dealing with exponents. To calculate exponents like 8^6, a standard calculator usually offers a specific button for the exponentiation process. This button may be labeled with a caret (^), 'exp', 'x^y', or something similar, depending on the calculator model.

To compute the exponent effectively, follow these steps:

1. Enter the base number (8 in our example).
2. Press the exponentiation button (^ or equivalent).
3. Type the exponent number (6).
4. Press 'equals' or 'enter' to get the result.

Modern calculators can handle large exponents, but it's essential to be aware of your calculator's limits, as extremely large powers can exceed the displayable range or the calculator's computational capacity. Additionally, understanding calculator syntax is critical, as the order of input can affect the calculation's outcome.

Mathematical Operations

Mathematical operations, such as addition, subtraction, multiplication, division, and exponentiation, are the building blocks of arithmetic. Each operation has specific rules and applications. For instance, exponentiation is a form of repeated multiplication, making it a powerful tool for expressing large numbers concisely or for performing calculations involving growth or decay.

It's essential to approach these operations methodically:

• Identify the operation type you are performing.
• Understand the properties of the operation (like the commutative property for addition).
• Follow the sequence of operations (remembering the order of operations: PEMDAS/BODMAS).

Mastering basic operations paves the way for solving more complex equations and understanding advanced mathematical concepts. Making a habit of solving operations step by step can prevent errors and foster a deeper comprehension of the problems at hand.