Problem 54 - College Algebra

Question

Simplify each exponential expression in Exercises 23–64. $$\frac{20 b^{10}}{10 b^{20}}$$

Step-by-Step Solution

Answer

The simplified form of given exponential expression \(\frac{20 b^{10}}{10 b^{20}}\) is \(2b^{-10}\)

Step 1: Simplify the Coefficients

First simplify fraction formed by the numeric coefficients. This is done by doing normal numerical division, which yields \( \frac{20}{10} = 2 \).

Step 2: Simplify the exponentials

To simplify the exponentials, we subtract the exponents of the like terms (the b's in this case) because the rule of exponent says when you divide like terms, you subtract their exponents. Here, we subtract the exponent of the divisor from the exponent of the dividend. Thus, \(b^{10-20} = b^{-10}\).

Step 3: Finalize Expression

Combine the answers from step 1 and step 2 to achieve the final simplified expression, which is \(2b^{-10}\).

Key Concepts

Division of Exponents

Dividing exponential expressions might seem tricky at first, but it's actually straightforward once you understand the rules. Whenever you divide powers with the same base, you must subtract the exponent of the denominator from the exponent of the numerator. This is known as the property of exponents.
For example, if you have a base like 'b', raised to two different powers, such as \(b^{m}/b^{n}\), you perform the operation \(b^{m-n}\).
This concept simplifies complex algebraic expressions significantly. By mastering this, you'll be able to handle any division of exponents within algebraic expressions smoothly. Just remember: when dividing, subtract the exponents as instructed, and only if the bases are the same.

Simplifying Fractions

Simplifying fractions involves reducing them to their lowest terms, where both the numerator and denominator share no common factors other than 1. When dealing with algebraic expressions, you follow slightly the same principle, but it often involves both numbers and variables.
Start by simplifying the numerical coefficients (the numbers in front of the variables). In the example \(\frac{20}{10}\), both numbers share a common factor '10'. Dividing both by 10 gives you 2, the simplest form of this fraction.
Next, apply the rules of exponents to simplify the expression involving variables, which often means subtracting the exponents as we've discussed. This approach helps maintain clarity and understanding in algebraic processes, and it's essential for solving algebraic expressions accurately.

Negative Exponents

A negative exponent indicates that the base is on the wrong side of a fraction and needs to be flipped over to become positive. The key rule to remember is \(a^{-n} = \frac{1}{a^{n}}\).
Negative exponents provide a method to express repeated division by a number more efficiently. In the expression \(b^{-10}\), it can be rewritten as \(\frac{1}{b^{10}}\).
It's essential to handle negative exponents correctly, as they often appear in simplified forms of expressions. Understanding the concept helps you in avoiding errors and in expressing the final algebraic expressions neatly.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations. These expressions form the foundation of algebra, allowing you to model real-world situations or solve various mathematical problems.
Expressions can look complex, but with the knowledge of exponent rules and simplification techniques, you can break them down into easily manageable parts. For example, \(2b^{-10}\) combines both a coefficient and a variable part that was simplified using exponent rules.
Understanding how to manipulate, simplify, and solve algebraic expressions is crucial, making problems easier and solutions more exact. Always remember that algebraic expressions, when mastered, offer great power in computational and real-world problem-solving.