Problem 48 - Algebra for College Students

Question

Simplify each expression. Express final results without using zero or negative integers as exponents. \(\left(b^{4}\right)^{-3}\)

Step-by-Step Solution

Answer

\(\frac{1}{b^{12}}\)

Step 1: Apply the Power of a Power Rule

Use the rule that says when you raise a power to another power, you multiply the exponents. This means for \(\left(b^4\right)^{-3}\), you multiply the exponents 4 and -3.

Step 2: Multiply the Exponents

Multiply the exponents: \(4 \times (-3) = -12\). This gives us \(b^{-12}\).

Step 3: Convert to Positive Exponents

Use the negative exponent rule, which states that \(a^{-b} = \frac{1}{a^b}\). Applying this, \(b^{-12} = \frac{1}{b^{12}}\).

Key Concepts

Understanding the Power of a Power Rule

When dealing with expressions involving exponents, sometimes you encounter a situation where one exponential expression is raised to another power. This is where the "power of a power rule" comes into play.

The power of a power rule helps us simplify such complex expressions. It states that you multiply the exponents when a number is raised to another exponential power.

• For example, \((b^4)^{-3}\) becomes \(b^{4 \times (-3)}\).

This rule simplifies the operation by changing the exponentiated bases into a single power expression. By multiplying the exponents, the expression becomes easier to handle in further simplification steps.

Multiplying Exponents Makes Complex Calculations Easy

Multiplying exponents can initially seem a bit confusing, but it's actually very straightforward when you understand the basic principles.

When you multiply two exponents in the context of the power of a power rule, this simply means you take the exponent of the base and multiply it by the exponent of the power.

• In our example, this involves multiplying 4 and -3, resulting in \(-12\).

This gives us a new expression, \(b^{-12}\), which still contains a negative exponent but is a critical step in simplifying the entire expression.

Mastering the Art of Simplifying Expressions

Simplifying expressions with exponents involves a few consistent steps to make them more manageable. Part of this process often includes converting negative exponents into positive ones to keep the expression straightforward and clear.

When dealing with negative exponents, remember the rule: a base raised to a negative exponent equals one over the base raised to the positive of that exponent.

• In mathematical terms, \(b^{-12}\) becomes \(\frac{1}{b^{12}}\).

Simplifying expressions into their simplest form not only makes them easier to understand but also more appropriate for use in subsequent mathematical operations.