Problem 59 - Elementary and Intermediate Algebra

Question

Simplify. Do not use negative exponents in the answer. \(b^{-7} \cdot b^{14}\)

Step-by-Step Solution

Answer

The simplified expression is \(b^7\).

Step 1: Identify the Base and Exponents

In this expression, we have the base as \(b\) with exponents \(-7\) and \(14\).

Step 2: Apply the Product of Powers Property

The product of powers property states that when multiplying two expressions with the same base, we add the exponents. Therefore, we have: \[ b^{-7} imes b^{14} = b^{-7 + 14} \]

Step 3: Simplify the Exponent

Simplify the exponent by performing the arithmetic operation: \[ b^{-7 + 14} = b^{7} \]

Step 4: Conclusion

The simplified form of the expression is \(b^7\), which no longer contains any negative exponents.

Key Concepts

Simplifying Expressions

Simplifying expressions is about making mathematical expressions easier to read and work with. When you simplify an expression, you essentially break it down into a much more straightforward form. Imagine having a recipe that uses complex measurements and you convert them into tablespoons and teaspoons. It's the same kind of idea here.

In algebra, simplification often involves combining like terms, canceling terms, or using certain mathematical rules to reduce the complexity. In our given example, the goal of the task is to simplify an expression with exponents:

• Identify the like terms
• Use exponent rules to combine them
• Ensure the final expression is free of negative exponents

By simplifying, you can make further calculations easier or more intuitive, helping to solve problems or equations much faster.

Product of Powers Property

The Product of Powers Property is a fundamental rule when working with exponents. It states that when you multiply two powers that have the same base, you should add their exponents together. This can be expressed as:

• If you have two terms like this: \( a^m \) and \( a^n \)
• Then their product will be: \( a^{m+n} \)

This principle is directly applied to our example: \( b^{-7} \cdot b^{14} \).

Here, both powers share the same base \(b\). According to the Product of Powers Property, you simply add their exponents: \(-7+14\), which simplifies to \(7\). So, the expression simplifies to \(b^7\).

The magic lies in its simplicity. Think of combining similar tasks to finish a project faster. This rule effectively shortens potentially cumbersome calculations.

Negative Exponents

Negative exponents might initially seem confusing, but they are really an invitation to look at numbers differently. A negative exponent indicates that you will take the reciprocal of the base. So, \( a^{-n} \) could be rewritten as \( \frac{1}{a^n} \).

In the context of simplifying expressions, the goal is often to eliminate negative exponents altogether. This results in a more standardized form of an expression. For instance, when you have powers like \( b^{-7} \), you’re being told that this is equivalent to \( \frac{1}{b^7} \). However, using rules like the Product of Powers Property, you often have shortcuts that eliminate the negatives altogether, such as turning \( b^{-7} \cdot b^{14} \) into \( b^{7} \) directly, without intermediary steps.

Handling negative exponents is crucial because it allows you to work with positive indices, which are significantly more intuitive and universally preferred in mathematical expressions.