Problem 10 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For Problems 10 through 15, factor \(b^{x}\) out of the following expressions. Check your answer by multiplying out. $$ b^{x+y}+b^{x} $$

Step-by-Step Solution

Answer

The factorization of \(b^{x+y}+b^{x}\) by taking \(b^{x}\) out is \(b^{x} (b^{y} + 1)\).

Step 1: Identifying the common factor

From the given expression \(b^{x+y}+b^{x}\), it can be noted that \(b^{x}\) is a common factor. The step now is to take this common factor \(b^{x}\) out of the expression.

Step 2: Applying the properties of exponents

Taking \(b^{x}\) common from \(b^{x+y}+b^{x}\) gives: \(b^{x} (b^{y} + 1)\). This is because \( b^{x} \times b^{y} = b^{x+y}\).

Step 3: Verification

To verify the factored expression, we can expand \( b^{x} (b^{y} + 1)\) again to get back the original expression. Expanding gives \(b^{x}\times b^{y} + b^{x} = b^{x+y} + b^{x}\). Hence, the verification is successful and our factorization is indeed correct.

Key Concepts

Common Factor

When faced with an algebraic expression like \( b^{x+y} + b^{x} \), one of the first steps in simplification through factorization is identifying the common factor. A common factor is a term that is present in all parts of the expression and can be factored out (removed by division) to simplify the expression. In our example, \( b^x \) is the term that appears in each part of the sum. By recognizing the common factor, one can rewrite the original expression in a more compact form.

In practical scenarios, spotting the common factor efficiently can make calculations quicker, especially as the complexity of the expressions increases. This also sets the stage for higher mathematics, where common factors play a crucial role in solving equations and manipulating formulas. It is, therefore, an essential skill to master in algebra.

Properties of Exponents

The process of factoring algebraic expressions heavily relies on understanding the properties of exponents. Exponents signify repeated multiplication, and these properties allow us to manipulate expressions involving powers in a consistent manner. When factoring out \(b^{x}\) from the expression \( b^{x+y} + b^{x} \), we apply one crucial property: multiplying powers with the same base.

Power of a Product Property

According to this property, \(b^{m} \times b^{n} = b^{m+n}\), when you multiply two exponents with the same base, you can simply add their powers. Our step-by-step solution uses this property to derive \(b^{x} (b^{y} + 1)\) from the original expression. Without a firm understanding of these properties, factorizing expressions can be challenging or even result in errors.

Verification of Factorization

An essential step after factorizing an expression is to verify the results. As mathematical expressions can become quite complex, ensuring the correctness of factored forms is necessary to prevent the propagation of errors in subsequent calculations. Verification is done by reversing the factorization process, which, in our example, involves expanding \( b^{x} (b^{y} + 1)\) to check if it equates to the original expression \( b^{x+y} + b^{x} \).

This step is akin to 'proofreading' in writing. It checks for the accuracy of the factorization, just like one would check a sentence for grammatical mistakes. Performing this verification builds confidence in the solution and reinforces understanding of the concepts involved in factoring. It also serves as good practice for validating solutions in more advanced mathematics where the factorization process can be a crucial step in problem-solving.