Problem 14 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Factor \(b^{x}\) out of the following expressions. Check your answer by multiplying out. $$ b^{3 x}-(2 b)^{-1+2 x} $$

Step-by-Step Solution

Answer

Factored form: \(b^{x}.\(b^{2x}-b^{-x}/2)\)

Step 1: Factor Out

Start by factoring out \(b^{x}\) from each term in the expression: \(b^{3x} - (2b)^{-1+2x}\) can be rewritten as \(b^{x} \cdot b^{2x} - b^{x} \cdot (2^{-1}b^{x})\)

Step 2: Simplify

Simplify the resulting expression: \(b^{x} (b^{2x} - 2^{-1}b^{x})\) = \(b^{x}(b^{2x}-b^{-x}/2)\)

Step 3: Check the Solution

Check if the factored form is correct by multiplying back the expressions: \(b^{x}(b^{2x}-b^{-x}/2)\) multiplied back gives \(b^{3x}-b^{x}/2\) which is simpliefied to \(b^{3x} - (2b)^{-1+2x}\), which is the original equation, confirming the factoring is correct

Key Concepts

Exponent Rules

Mastering exponent rules is crucial when dealing with algebraic expressions involving powers. A solid understanding of these rules facilitates the simplification of complex expressions and plays a major role in various areas of mathematics.

Basic exponent rules include:

• Product of Powers: When multiplying powers with the same base, you add the exponents, such as in \( b^m \times b^n = b^{m+n} \).
• Quotient of Powers: When dividing powers with the same base, you subtract the exponents, like \( b^m \div b^n = b^{m-n} \).
• Power of a Power: When raising a power to another power, you multiply the exponents, demonstrated as \( (b^m)^n = b^{m \times n} \).
• Negative Exponent: Indicates reciprocal, which means \( b^{-n} = 1\/b^n \).

Apply these rules correctly, and you can transform expressions to more manageable forms or factor common terms as seen in the provided exercise. It's the manipulation of exponents according to these rules that enables the factoring and simplifying process.

Factoring Out Common Factors

Factoring out common factors is a technique used to simplify expressions by identifying and removing factors common to all terms. This core concept is often a first step in solving complex equations and simplifying algebraic expressions.

To factor out a common factor:

• Identify the smallest exponent for the common base in the terms.
• Use this as your common factor to remove from each term.
• Divide each term by this common factor and write it separated from the remaining expression.

In our exercise, we note that \(b^x\) is the common factor in each term in the expression \(b^{3x} - (2b)^{-1+2x}\). By using the smallest exponent of the variable \(b\), which is \(x\), we factor out \(b^x\) from each term, simplifying the process of manipulation that follows.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is all about expressing equations in their simplest form, making it easier to understand and solve them. This process usually involves factoring, combining like terms, and applying exponent rules.

For simplification, follow these guidelines:

• Combine like terms, which are terms with the same variable raised to the same power.
• Reduce fractions and simplify any numerical coefficients.
• Apply exponent rules to simplify terms with variables raised to powers.

As shown in the solution steps, after factoring out \(b^x\), the expression is simplified by applying the exponent and coefficient rules, resulting in \(b^x(b^{2x}-b^{-x}/2)\). Simplifying algebraic expressions not only leads to more concise solutions but also makes it possible to more easily identify further steps necessary for fully solving algebraic problems.