Problem 32 - Fundamentals of Mathematics

Question

Determine the value of each of the powers. Use a calculator to check each result. \(1^{2}\)

Step-by-Step Solution

Answer

The value of \(1^2\) is 1.

Step 1: Understand the Problem

Identify the power expression that needs to be evaluated: \(1^2\). This means we have 1 raised to the power of 2.

Step 2: Evaluate the Power

Use the property of exponents: any number raised to the power of 2 means that it's multiplied by itself one time. Therefore, \(1^2 = 1 \times 1\).

Step 3: Calculate the Product

Calculate the product of multiplying 1 by itself: \(1 \times 1 = 1\).

Step 4: Verify with a Calculator

Use a calculator to verify the result of \(1^2\). Enter the calculation into a calculator to confirm that the result is indeed 1.

Key Concepts

Exponentiation

Exponentiation is a fundamental mathematical operation that involves raising a number, known as the base, to the power of another number, called the exponent. It describes how many times the base is multiplied by itself. In the example given, the expression is \(1^2\). Here,

• 1 is the base.
• 2 is the exponent.

The exponent tells us to multiply the base by itself once. This is written as:\[1^2 = 1 \times 1\]This operation is straightforward because no matter how many times you multiply 1 by itself, the result is always 1. Exponentiation becomes more interesting and complex when working with other numbers, especially larger ones. It is a powerful tool used extensively in various fields of science and engineering.

Using a Calculator for Math Problems

Using a calculator can greatly simplify the process of solving math problems, especially when dealing with exponents. While simple expressions like \(1^2\) are easy to do manually, calculators can save time and reduce errors in more complex calculations. Here's why calculators are beneficial:

• They provide quick and accurate results.
• They are user-friendly and can handle complex mathematical functions.
• They assist in visualizing large numbers and results.

To verify the result of \(1^2\) using a calculator, simply enter the base, then use the exponentiation function (often marked as \(x^y\) or \(^\ddagger\)) followed by the exponent. For more elaborate expressions, double-checking your calculations using a calculator increases confidence in your answer.

Properties of Exponents

Understanding the properties of exponents is crucial for simplifying expressions and solving problems efficiently. Here are some key properties to remember:

• **Product of Powers:** \(a^m \times a^n = a^{m+n}\)
• **Power of a Power:** \((a^m)^n = a^{m \times n}\)
• **Power of a Product:** \((ab)^n = a^n \times b^n\)
• **Zero Exponent:** Any non-zero number raised to the power of 0 is 1, i.e., \(a^0 = 1\) for \(a eq 0\).

These properties make it much easier to work with exponential expressions and simplify them. For example, applying the zero exponent rule, we know that \(5^0 = 1\) regardless of what the base is, as long as it is not zero. These rules are essential tools for high school algebra and beyond. Knowing when and how to implement these rules helps solve exponential problems without confusion.