Problem 90 - Algebra and Trigonometry Real Mathematics, Real People

Question

Think About It In Exercises \(89-92,\) place the correct symbol \(( < \text { or } > )\) between the two numbers. $$2^{10} \quad 10^{2}$$

Step-by-Step Solution

Answer

\(2^{10} > 10^{2}\)

Step 1: Calculate \(2^{10}\)

To begin, calculate \(2^{10}\), which means multiplying 2 by itself 10 times. This calculation results in 1024.

Step 2: Calculate \(10^{2}\)

Next, calculate \(10^{2}\), which means multiplying 10 by itself 2 times. This calculation gives us 100.

Step 3: Compare the Results

Finally, compare the results of the two calculations. Since 1024 is greater than 100, the correct symbol to place between \(2^{10}\) and \(10^{2}\) is `>`.

Key Concepts

Comparison of Numbers

When comparing numbers, we use symbols like `<` for "less than" and `>` for "greater than" to indicate the relationship between two values. This concept helps us understand which of the two numbers is bigger or smaller. For the exercise given, comparing the values of two expressions, \(2^{10}\) and \(10^{2}\), helps determine which expression results in a larger number.

• Start by calculating each number separately.
• Once you have both numerical values, you simply check to see which number is larger or smaller.
• Inserting the correct symbol, like `<` or `>`, based on this comparison displays the relationship clearly.

In our example, \(2^{10} = 1024\) and \(10^{2} = 100\). Since 1024 is greater than 100, the proper symbol to use is `>`. Understanding how to make these comparisons is fundamental in mathematics, permitting straightforward evaluations of not only numbers but also complex mathematical expressions.

Powers of Numbers

The concept of powers, or exponentiation, describes when a number is multiplied by itself a certain number of times. The power of a number is denoted as \(a^{b}\), where \(a\) is the base and \(b\) is the exponent.

• The base is the number being multiplied.
• The exponent indicates how many times the base is used as a factor.
• For example, \(2^{10}\) involves multiplying 2 by itself 10 times, resulting in 1024.
• Similarly, \(10^{2}\) means multiplying 10 by itself 2 times, resulting in 100.

Understanding this operation is essential because powers grow rapidly and can be used to simplify expressions.
Powers are crucial not only for identifying how large or small a number is but also for doing quick calculations in various fields such as physics, engineering, and computer science. Moreover, knowing how to calculate powers of numbers gives us the tools necessary to handle mathematical problems efficiently.

Mathematical Symbols

Mathematical symbols are the shorthand languages of numbers and operations, allowing us to express complex ideas in a succinct way. Some common symbols used for comparison include:

• `<` (less than)
• `>` (greater than)
• `=` (equals)

These symbols are universal tools for communicating mathematical relationships.
When we write \(2^{10} > 10^{2}\), it conveys that 1024 is greater than 100 without needing additional words. This universal language makes it easier to compare and solve problems in math.
Notably, symbols help in various areas such as algebra, calculus, and statistics, where clarity and precision are paramount. Recognizing and mastering these symbols can enhance problem-solving skills and improve mathematical literacy significantly. Understanding and utilizing these symbols appropriately is key to efficient mathematical communication and analysis.