Problem 24 - Beginning and Intermediate Algebra

Question

Simplify. $$\left(-6 b^{-4}\right)\left(2 b^{-7}\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(-12b^{-11}\).

Step 1: Write the given expression

The given expression is: \[ \left(-6 b^{-4}\right)\left(2 b^{-7}\right) \]

Step 2: Multiply the constant terms

Multiply the constants -6 and 2: \[ (-6) * (2) = -12 \]

Step 3: Apply the rules of exponents to multiply the exponential terms

When multiplying exponential terms with the same base, we add the exponents: \[ b^{-4} * b^{-7} = b^{-4 + (-7)} = b^{-11} \]

Step 4: Combine the constant and exponential terms

Now, we combine the constant term and exponential term to obtain the final simplified expression: \[ \underbrace{-12}_{\text{Constant term}}\ \underbrace{b^{-11}}_{\text{Exponential term}} \]

Step 5: Write the simplified expression

The simplified expression is: \[ -12b^{-11} \]

Key Concepts

Simplification of Expressions

Simplification of expressions is an essential skill in algebra. It involves reducing an expression into its simplest form. This process makes expressions easier to understand, compare, or solve. In our example, simplification involves working with both numbers and variables.

Here, we deal with constants and variable terms:

• The constants in the expression are the numbers, like -6 and 2.
• The variable part includes the base, represented as \(b\), raised to some power.

The goal in simplification is to combine like terms. To simplify complex expressions effectively, understanding fundamental algebraic principles and methods is crucial.

Multiplying Exponential Terms

When you're multiplying exponential terms, it's important to focus on the bases and exponents. The base is the repeated factor, and the exponent tells us how many times to multiply that base by itself. In our specific example, each term of the variable is raised to a power.

To multiply exponential terms with the same base:

• Identify the base. In our expression, it's \(b\).
• Add the exponents together. In this exercise, we have \(b^{-4}\) and \(b^{-7}\): the rule is \(a^m \cdot a^n = a^{m+n}\), so \(b^{-4} \cdot b^{-7} = b^{-11}\).

Understanding this is vital in algebra, as it simplifies expressions involving exponents by reducing them into a single power expression.

Rules of Exponents

The rules of exponents are a set of guidelines that dictate how to handle expressions involving powers. These rules are fundamental in simplifying and working with expressions efficiently. Let's break down some major rules:

1. **Product of Powers Rule**: This is the rule we used in our exercise. When multiplying like bases, add the exponents: \(a^m \cdot a^n = a^{m+n}\).2. **Negative Exponents**: An exponent with a negative sign indicates the reciprocal of that base raised to the opposite positive power: \(a^{-n} = \frac{1}{a^n}\).Applying negative exponents correctly is crucial in mathematics. In our exercise, the base \(b\) had negative exponents (-4, -7). We added them up to get \(-11\) as the new exponent.

Comprehending these rules helps significantly in simplifying different algebraic expressions and controlling the number of expressions.