Problem 16 - Beginning and Intermediate Algebra

Question

Simplify. $$b^{-4} \cdot b^{12}$$

Step-by-Step Solution

Answer

The simplified expression is \(b^8\).

Step 1: Identify the base and exponents

In the given expression, \(b^{-4} \cdot b^{12}\), the base is \(b\), and the exponents are \(-4\) and \(12\).

Step 2: Apply the property of exponents

Since we have the same base, we will add the exponents as per the property of exponents: \(a^m \cdot a^n = a^{m+n}\). In our case, we have: \(b^{-4} \cdot b^{12} = b^{-4 + 12}\)

Step 3: Add the exponents

Now, let's add the exponents \(-4\) and \(12\): \(-4 + 12 = 8\) So, our expression becomes: \(b^{-4} \cdot b^{12} = b^8\)

Step 4: Write the simplified expression

We have successfully simplified the expression: \(b^{-4} \cdot b^{12} = b^8\)

Key Concepts

Understanding Negative Exponents

Negative exponents might seem tricky at first, but they actually follow a simple rule. An exponent indicates how many times the base is multiplied by itself. A negative exponent, like \( b^{-4} \), tells us to do the reciprocal of the base raised to the positive exponent.

• The expression \( b^{-4} \) can be rewritten as \( \frac{1}{b^4} \).
• This does not mean that the base is negative, but rather that it's the reciprocal.

Understanding this concept helps when multiplying or dividing expressions with negative exponents. In practice, when we multiply bases with exponents, as long as the bases are the same, we add the exponents even if they are negative. This is what makes them simple and yet vital in simplifying expressions.

Simplifying Expressions with Exponents

Simplifying expressions is crucial in algebra to make calculations more straightforward. When you have expressions like \( b^{-4} \cdot b^{12} \), it is important to identify common bases and work with the exponent rules. Here are the basics:

• Identify the base: Both operands should have the same base, here it's \( b \).
• Apply the addition of exponents: When multiplying, you add the exponents, \( a^m \cdot a^n = a^{m+n} \).
• Simplified example: \( b^{-4} \cdot b^{12} = b^{-4+12} = b^8 \).

These steps ensure that algebraic operations are easier to manage, paving the way for solving more complex problems.

Algebraic Expressions and Their Manipulation

Algebraic expressions are combinations of numbers, variables, and operations (like addition and multiplication). They are the building blocks of mathematical equations. Here's how they are handled:

• Variables can be raised to any power, including negative ones, like \( b^{-4} \).
• The goal often is to simplify or manipulate these expressions to solve equations.
• Using rules like combining like terms or using distributive properties can simplify expressions.

Working with algebraic expressions involves recognizing patterns and applying appropriate rules, like handling negative exponents or multiplying terms with the same base efficiently. Being comfortable with these concepts helps us deal with more complicated algebraic problems.