Problem 76 - College Algebra with Modeling and Visualization

Question

Exercises \(65-90:\) Use rules of exponents to simplify the expression. Use positive exponents to write your answer. $$ \frac{\left(b^{2}\right)^{-1}}{\left(b^{-4}\right)^{3}} $$

Step-by-Step Solution

Answer

The simplified expression is \( b^{10} \).

Step 1: Simplify the Numerator

The numerator of the expression is \( (b^2)^{-1} \). According to the exponent rule \((x^m)^{-n} = x^{-mn}\), we can write this as \( b^{-2} \).

Step 2: Simplify the Denominator

The denominator \((b^{-4})^3\) can be simplified using the power of a power rule \((x^m)^n = x^{mn}\) to get \( b^{-12} \).

Step 3: Divide by Subtracting Exponents

We now have the expression \( \frac{b^{-2}}{b^{-12}} \). We use the quotient rule for exponents \(\frac{x^a}{x^b} = x^{a-b}\), giving us \( b^{-2 - (-12)} = b^{10} \).

Step 4: Ensure Positive Exponents

The expression \( b^{10} \) is already in terms of positive exponents, so it becomes our final simplified form.

Key Concepts

Simplifying Expressions

Understanding how to simplify expressions is an important skill in algebra. Simplification involves rewriting an expression in a way that makes it easier to work with or understand. In this exercise, we start by simplifying the individual parts of a fraction.The expression given is \( \frac{\left(b^{2}\right)^{-1}}{\left(b^{-4}\right)^{3}} \). We first focus on simplifying the numerator and the denominator separately. By applying the exponent rules correctly, we can rewrite them in a simpler form before combining them into a final expression.Simplifying involves transforming complex powers into more manageable numbers. Once the individual components are simplified, you can then combine or further transform the expression as required, often using the properties of numbers to achieve this.

Rules of Exponents

The rules of exponents are essential tools for manipulating and simplifying expressions involving powers. They allow us to rewrite exponential expressions in different forms and solve complex mathematical problems.Let's explore some of the key exponent rules:

• Power of a Power: When raising a power to another power, multiply the exponents: \((x^m)^n = x^{m \cdot n}\).
• Negative Exponent Rule: A negative exponent indicates taking the reciprocal of the base raised to a positive exponent: \(x^{-n} = \frac{1}{x^n}\).
• Quotient Rule: When dividing like bases, subtract the exponents: \(\frac{x^a}{x^b} = x^{a-b}\).

In our exercise, we use these rules to simplify the numerator and denominator, transforming the expression into a single term with a positive exponent.

Positive Exponents

Working with positive exponents ensures that our final expression is easier to interpret. Positive exponents show how many times a number (the base) is multiplied by itself.When simplifying or rearranging expressions, always aim to convert all terms to have positive exponents. This often involves using the negative exponent rule to flip the base. For example, \(b^{-2}\) becomes \(\frac{1}{b^2}\) when expressed as a positive exponent.In this exercise, at the final step, we ended up with \(b^{10}\), which is a simplified form already expressed in positive exponents. This means our solution is neat and ready to use without further transformation. Positive exponents provide clarity by straightforwardly showing repeated multiplication, which helps with further calculations and interpretations.