Problem 54 - College Algebra

Question

Simplify each exponential expression $$ \frac{20 b^{10}}{10 b^{20}} $$

Step-by-Step Solution

Answer

The simplified exponential expression is \(2b^{-10}\).

Step 1: Simplify the Constants

Divide the constant 20 by 10 to get 2.

Step 2: Simplify the Variables With Exponents

Apply the rule of exponents. Subtract the exponent of \(b\) in the denominator from the exponent of \(b\) in the numerator; i.e., \(b^{10 - 20}\).

Step 3: Write the final simplified expression

Combine the simplified constants and the simplified variable with exponent to get the final expression - \(2b^{-10}\).

Key Concepts

Exponential Expressions

When we talk about exponential expressions, we refer to mathematical expressions that involve exponents. In an exponential expression, a base number is raised to a power. This power, or exponent, indicates how many times the base number is multiplied by itself. For example, in the expression \(b^{10}\), \(b\) is the base, and \(10\) is the exponent. This means \(b\) is multiplied by itself 10 times.

• The base tells you what number is repeated as a factor.
• The exponent tells you how many times the base is used as a factor.

Exponential expressions are useful because they offer a shorthand way to represent very large or very small numbers. Instead of writing \(b \times b \times b \times \ldots\), we simply write \(b^{10}\). This format makes mathematical operations easier to perform and understand.

Exponent Rules

Knowing the exponent rules is crucial for simplifying exponential expressions. These rules make it easier to manipulate expressions with exponents without expanding them. Let's explore some important rules:

• **Product Rule**: If you multiply two exponential expressions with the same base, you add the exponents. For example, \(b^m \times b^n = b^{m+n}\).
• **Quotient Rule**: When dividing exponential expressions with the same base, subtract the exponent in the denominator from the exponent in the numerator: \(\frac{b^m}{b^n} = b^{m-n}\).
• **Power Rule**: When raising an exponential expression to another power, multiply the exponents: \((b^m)^n = b^{mn}\).

In our problem, \(\frac{b^{10}}{b^{20}}\), we apply the quotient rule to obtain \(b^{10 - 20}\), which simplifies to \(b^{-10}\). Knowing these exponent rules streamlines the process of working with exponential terms.

Numerator and Denominator Simplification

Simplifying the numerator and denominator separately is a methodical technique used to simplify fractional exponential expressions. This approach ensures clarity and precision in reducing expressions to their simplest forms. Let's break down the process:

• **Simplify the Constants**: Before dealing with variables and exponents, start by simplifying the coefficients, or numbers, in both the numerator and the denominator. For instance, \(\frac{20}{10} = 2\).
• **Simplify the Variables**: Next, apply the quotient rule from our exponent rules to the variables with exponents. In our case, we simplify \(\frac{b^{10}}{b^{20}}\) to \(b^{-10}\) by subtracting the exponents \(10 - 20\).

Once both parts are simplified, combine them into a single simplified expression. For the given exercise, the result is \(2b^{-10}\). Following these steps ensures the exponential expression is fully simplified, making it easier to understand and work with.