Simplifying Exponential Expressions A Step By Step Guide

In the realm of mathematics, simplifying expressions is a fundamental skill, and when dealing with exponents, it becomes an art form. This article delves into the process of simplifying a specific exponential expression, providing a detailed, step-by-step guide to not only arrive at the solution but also understand the underlying principles. Let's embark on this journey of simplification, where we transform a seemingly complex expression into a concise and elegant form.

The Expression at Hand

Our mission is to simplify the following expression:

$$\frac{b^{\frac{1}{2}} b^{-\frac{4}{3}}}{b^{\frac{1}{3}}}$$

This expression involves the variable 'b' raised to various fractional exponents. Our goal is to manipulate this expression using the rules of exponents to arrive at a simplified form, where 'b' is raised to a single positive exponent. This simplification process not only makes the expression easier to understand but also facilitates further mathematical operations.

Step 1: Product of Powers in the Numerator

The first step in simplifying this expression involves dealing with the numerator. We have the product of two terms, both involving 'b' raised to different exponents: $b^{\frac{1}{2}}$ and $b^{-\frac{4}{3}}$.

The key rule of exponents that comes into play here is the product of powers rule: When multiplying powers with the same base, you add the exponents. Mathematically, this is expressed as:

$$x^m \cdot x^n = x^{m+n}$$

Applying this rule to our numerator, we add the exponents $\frac{1}{2}$ and $-\frac{4}{3}$:

$$\frac{1}{2} + \left(-\frac{4}{3}\right) = \frac{1}{2} - \frac{4}{3}$$

To add these fractions, we need a common denominator. The least common multiple of 2 and 3 is 6. So, we convert both fractions to have a denominator of 6:

$$\frac{1}{2} = \frac{1 \cdot 3}{2 \cdot 3} = \frac{3}{6}$$

$$\frac{4}{3} = \frac{4 \cdot 2}{3 \cdot 2} = \frac{8}{6}$$

Now we can subtract the fractions:

$$\frac{3}{6} - \frac{8}{6} = \frac{3 - 8}{6} = \frac{-5}{6}$$

Therefore, the numerator simplifies to:

$$b^{\frac{1}{2}} b^{-\frac{4}{3}} = b^{-\frac{5}{6}}$$

This step is crucial as it combines the two terms in the numerator into a single term with a single exponent. This makes the expression more manageable and sets the stage for the next step in the simplification process. Remember, the product of powers rule is a cornerstone of exponent manipulation, and understanding its application is essential for simplifying complex expressions.

Step 2: Quotient of Powers

Now that we've simplified the numerator, our expression looks like this:

$$\frac{b^{-\frac{5}{6}}}{b^{\frac{1}{3}}}$$

This is a fraction where both the numerator and denominator have the same base, 'b', raised to different exponents. To simplify this, we'll use another fundamental rule of exponents: the quotient of powers rule.

The quotient of powers rule states that when dividing powers with the same base, you subtract the exponent in the denominator from the exponent in the numerator. Mathematically, this is expressed as:

$$\frac{x^m}{x^n} = x^{m-n}$$

Applying this rule to our expression, we subtract the exponent in the denominator, $\frac{1}{3}$, from the exponent in the numerator, $-\frac{5}{6}$:

-rac{5}{6} - \frac{1}{3}

Again, we need a common denominator to subtract these fractions. The least common multiple of 6 and 3 is 6. We convert $\frac{1}{3}$ to have a denominator of 6:

$$\frac{1}{3} = \frac{1 \cdot 2}{3 \cdot 2} = \frac{2}{6}$$

Now we can perform the subtraction:

-rac{5}{6} - \frac{2}{6} = \frac{-5 - 2}{6} = \frac{-7}{6}

Therefore, the expression simplifies to:

$$\frac{b^{-\frac{5}{6}}}{b^{\frac{1}{3}}} = b^{-\frac{7}{6}}$$

This step effectively reduces the fraction to a single term with a single exponent. The quotient of powers rule is instrumental in simplifying expressions involving division of powers with the same base. By subtracting the exponents, we consolidate the expression and move closer to our final simplified form.

Step 3: Dealing with Negative Exponents

At this point, our expression is:

$$b^{-\frac{7}{6}}$$

However, the problem statement specifies that our final answer should only contain positive exponents. We currently have a negative exponent, $-\frac{7}{6}$. To address this, we'll use the rule for negative exponents.

The negative exponent rule states that a term raised to a negative exponent is equal to the reciprocal of that term raised to the positive version of the exponent. Mathematically, this is expressed as:

$$x^{-n} = \frac{1}{x^n}$$

Applying this rule to our expression, we get:

$$b^{-\frac{7}{6}} = \frac{1}{b^{\frac{7}{6}}}$$

This transformation effectively removes the negative exponent and expresses the term as a reciprocal with a positive exponent. Understanding and applying the negative exponent rule is crucial for adhering to the problem's constraints and expressing the final answer in the desired format.

Final Simplified Expression

After applying all the necessary rules of exponents, we have arrived at the final simplified expression:

$$\frac{1}{b^{\frac{7}{6}}}$$

This expression represents the simplified form of the original expression, adhering to the requirement of using only positive exponents. By systematically applying the product of powers rule, the quotient of powers rule, and the negative exponent rule, we have successfully transformed a complex expression into a more manageable and understandable form.

Summary of Steps

To recap, here's a step-by-step summary of how we simplified the expression:

1. Product of Powers in the Numerator: Applied the rule $x^m \cdot x^n = x^{m+n}$ to simplify the numerator.
2. Quotient of Powers: Applied the rule $\frac{x^m}{x^n} = x^{m-n}$ to simplify the fraction.
3. Dealing with Negative Exponents: Applied the rule $x^{-n} = \frac{1}{x^n}$ to eliminate the negative exponent.

By mastering these rules and applying them strategically, you can confidently simplify a wide range of exponential expressions. Remember, practice is key to developing fluency in these techniques.

Key Takeaways for Simplifying Exponential Expressions

Simplifying exponential expressions can seem daunting at first, but by understanding and applying a few key rules, you can break down complex problems into manageable steps. Let's reinforce those takeaways:

• Master the Rules: The product of powers rule ($x^m \cdot x^n = x^{m+n}$), the quotient of powers rule ($\frac{x^m}{x^n} = x^{m-n}$), and the negative exponent rule ($x^{-n} = \frac{1}{x^n}$) are your primary tools. Know them inside and out.
• Work Step-by-Step: Don't try to do everything at once. Break the problem down into smaller steps, simplifying one part at a time. This reduces the chance of errors and makes the process clearer.
• Find Common Denominators: When adding or subtracting exponents that are fractions, you'll need a common denominator. Take the time to find it accurately.
• Pay Attention to the Instructions: Always double-check the problem statement for any specific requirements, such as using only positive exponents in the final answer.
• Practice Regularly: Like any mathematical skill, simplifying exponential expressions gets easier with practice. Work through various examples to build your confidence and speed.

By keeping these key takeaways in mind, you'll be well-equipped to tackle any exponential expression that comes your way. The journey of simplification is a rewarding one, leading to a deeper understanding of mathematical principles and enhanced problem-solving abilities. Remember, each simplified expression is a testament to your growing mathematical prowess.