Problem 45 - Algebra 2

Question

CHALLENGE Determine which is greater, \(100^{10}\) or \(10^{100} .\) Explain.

Step-by-Step Solution

Answer

\(10^{100}\) is greater than \(100^{10}\).

Step 1: Understanding the exponential expressions

We need to compare two large numbers given in exponential form: \(100^{10}\) and \(10^{100}\). These expressions represent 100 raised to the power of 10 and 10 raised to the power of 100, respectively.

Step 2: Rewriting the bases

Notice that the base of the first expression, 100, can be rewritten as \(10^2\). So, \(100^{10}\) is equivalent to \((10^2)^{10}\).

Step 3: Applying exponent rules

Using the rule \((a^m)^n = a^{m \cdot n}\), we can simplify \((10^2)^{10}\) to \(10^{2 \times 10} = 10^{20}\). Hence, \(100^{10} = 10^{20}\).

Step 4: Comparing the expressions

Now we have \(100^{10}\) rewritten as \(10^{20}\) and need to compare this to \(10^{100}\). It is clear that these are both powers of 10.

Step 5: Final comparison

When comparing exponential expressions with the same base, the expression with the larger exponent is greater. Since \(20 < 100\), we conclude that \(10^{20} < 10^{100}\). Therefore, \(100^{10} < 10^{100}\).

Key Concepts

Exponent Rules

Exponential expressions can initially appear daunting, especially when dealing with very large numbers or calculations. Thankfully, exponent rules offer a way to simplify and manage such expressions. Here are some key rules to remember:

• Product of Powers: When you multiply like bases, you add the exponents. For example, \(a^m \times a^n = a^{m+n}\).
• Power of a Power: When you raise a power to another power, you multiply the exponents. For instance, \((a^m)^n = a^{m \cdot n}\).
• Power of a Product: When a product is raised to an exponent, you can distribute the exponent: \((ab)^m = a^m \times b^m\).

By applying the Power of a Power rule, the expression \((10^2)^{10}\) can be transformed to \(10^{20}\), making it simpler to handle. These rules are foundational for managing complex exponential expressions efficiently.

Comparing Exponential Expressions

Comparing exponential expressions involves analyzing the base and the exponent. This is critical when determining the relative size of numbers written in exponential form. With the problem at hand, we have two expressions:

• \(100^{10}\)
• \(10^{100}\)

Initially, these appear unrelated. However, by recognizing that 100 is equivalent to \(10^2\), we can rewrite \(100^{10}\) as \((10^2)^{10}\) and further simplify it to \(10^{20}\) using exponent rules.
Once both expressions have the same base, comparing them becomes straightforward. Exponential expressions with identical bases can be easily compared by examining the exponents. The larger exponent indicates the larger quantity. Here, \(10^{20}\) is compared with \(10^{100}\), where 100 is the larger exponent.

Simplifying Exponents

Simplifying exponents is a crucial skill in mathematics that reduces complex expressions to more manageable forms. The aim is to use exponent rules to rewrite expressions for easier interpretation or comparison. In our example, the expression \(100^{10}\) was simplified to \(10^{20}\) by recognizing and exploiting the relation \(100 = 10^2\).
This simplification involves two main steps:

• Rewriting the Base: Recognize that 100 is \(10^2\) so that \(100^{10}\) becomes \((10^2)^{10}\).
• Applying the Power of a Power Rule: Use \((a^m)^n = a^{m \cdot n}\) to simplify to \(10^{20}\).

By reducing to a simpler form, we can more easily understand and compare exponential expressions, illustrating the significant power of simplifying exponents in solving mathematical challenges.