Simplifying Exponential Expressions 2⁴ * 9⁻² * 5⁻³ Divided By 8 * 3⁻⁵ * 125³ A Step-by-Step Guide

Simplifying exponential expressions can seem daunting at first, but with a systematic approach and a solid understanding of the rules of exponents, it becomes a manageable and even enjoyable task. In this comprehensive guide, we'll break down the process of simplifying a complex exponential expression, 2⁴ * 9⁻² * 5⁻³ Divided by 8 * 3⁻⁵ * 125³, step by step. We'll cover the fundamental rules of exponents, demonstrate how to apply them effectively, and provide helpful tips and tricks along the way. So, let's dive in and master the art of simplifying exponential expressions!

Understanding the Fundamentals of Exponential Expressions

Before we tackle the main problem, it's crucial to have a firm grasp of the basic concepts and rules of exponents. Exponential expressions consist of two main components: the base and the exponent. The base is the number being multiplied, and the exponent indicates how many times the base is multiplied by itself. For instance, in the expression 2⁴, 2 is the base, and 4 is the exponent, meaning 2 is multiplied by itself four times (2 * 2 * 2 * 2 = 16).

The rules of exponents are the key to simplifying these expressions. Let's review some of the most important ones:

1. Product of Powers Rule: When multiplying exponential expressions with the same base, you add the exponents. Mathematically, this is expressed as: aᵐ * aⁿ = aᵐ⁺ⁿ. For example, 2² * 2³ = 2²⁺³ = 2⁵ = 32.
2. Quotient of Powers Rule: When dividing exponential expressions with the same base, you subtract the exponents. This rule is represented as: aᵐ / aⁿ = aᵐ⁻ⁿ. For example, 3⁵ / 3² = 3⁵⁻² = 3³ = 27.
3. Power of a Power Rule: When raising an exponential expression to another power, you multiply the exponents. This rule is written as: (aᵐ)ⁿ = aᵐⁿ. For example, (2²)³ = 2²*³ = 2⁶ = 64.
4. Power of a Product Rule: When raising a product to a power, you distribute the power to each factor in the product. This rule is expressed as: (ab)ⁿ = aⁿbⁿ. For example, (2 * 3)² = 2² * 3² = 4 * 9 = 36.
5. Power of a Quotient Rule: When raising a quotient to a power, you distribute the power to both the numerator and the denominator. This rule is written as: (a/b)ⁿ = aⁿ/bⁿ. For example, (4/2)³ = 4³/2³ = 64/8 = 8.
6. Negative Exponent Rule: A negative exponent indicates the reciprocal of the base raised to the positive value of the exponent. This rule is expressed as: a⁻ⁿ = 1/aⁿ. For example, 2⁻² = 1/2² = 1/4.
7. Zero Exponent Rule: Any non-zero number raised to the power of zero equals 1. This rule is written as: a⁰ = 1 (where a ≠ 0). For example, 5⁰ = 1.

With these rules in mind, we're well-equipped to tackle our main problem.

Step-by-Step Simplification of 2⁴ * 9⁻² * 5⁻³ Divided by 8 * 3⁻⁵ * 125³

Now, let's break down the simplification of the expression 2⁴ * 9⁻² * 5⁻³ Divided by 8 * 3⁻⁵ * 125³ step by step. Our goal is to express the entire expression using the simplest possible terms and positive exponents.

Step 1: Rewrite all numbers as products of their prime factors.

This step is crucial because it allows us to apply the rules of exponents more effectively. We need to express each number in the expression as a product of its prime factors. Recall that a prime number is a whole number greater than 1 that has only two divisors: 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, and so on.

• 2⁴ remains as it is since 2 is a prime number.
• 9 can be expressed as 3², so 9⁻² becomes (3²)⁻².
• 5⁻³ remains as it is since 5 is a prime number.
• 8 can be expressed as 2³, so the expression becomes 2³.
• 3⁻⁵ remains as it is since 3 is a prime number.
• 125 can be expressed as 5³, so 125³ becomes (5³)³.

Our expression now looks like this: 2⁴ * (3²)⁻² * 5⁻³ Divided by 2³ * 3⁻⁵ * (5³)³.

Step 2: Apply the Power of a Power Rule.

Next, we'll apply the power of a power rule, which states that (aᵐ)ⁿ = aᵐⁿ. This means we'll multiply the exponents where a power is raised to another power.

• (3²)⁻² becomes 3²*⁻² = 3⁻⁴.
• (5³)³ becomes 5³*³ = 5⁹.

Our expression now becomes: 2⁴ * 3⁻⁴ * 5⁻³ Divided by 2³ * 3⁻⁵ * 5⁹.

Step 3: Rewrite the division as a fraction.

To make it easier to apply the quotient of powers rule, let's rewrite the division as a fraction:

(2⁴ * 3⁻⁴ * 5⁻³) / (2³ * 3⁻⁵ * 5⁹)

Step 4: Apply the Quotient of Powers Rule.

Now, we can apply the quotient of powers rule (aᵐ / aⁿ = aᵐ⁻ⁿ) to simplify the expression further. We'll apply this rule separately to the terms with the same base (2, 3, and 5).

• For the base 2: 2⁴ / 2³ = 2⁴⁻³ = 2¹ = 2.
• For the base 3: 3⁻⁴ / 3⁻⁵ = 3⁻⁴⁻⁽⁻⁵⁾ = 3⁻⁴⁺⁵ = 3¹ = 3.
• For the base 5: 5⁻³ / 5⁹ = 5⁻³⁻⁹ = 5⁻¹².

Our expression now simplifies to: 2 * 3 * 5⁻¹².

Step 5: Eliminate the negative exponent.

To eliminate the negative exponent, we'll use the negative exponent rule (a⁻ⁿ = 1/aⁿ). This means we'll move the term with the negative exponent to the denominator and change the exponent to positive.

5⁻¹² becomes 1/5¹².

Our expression now becomes: (2 * 3) / 5¹².

Step 6: Simplify the numerator.

Finally, let's simplify the numerator by multiplying 2 and 3:

2 * 3 = 6.

Final Simplified Expression:

Therefore, the simplified form of the expression 2⁴ * 9⁻² * 5⁻³ Divided by 8 * 3⁻⁵ * 125³ is 6 / 5¹².

Common Mistakes to Avoid

When simplifying exponential expressions, it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

• Forgetting the order of operations: Always remember to follow the order of operations (PEMDAS/BODMAS) when simplifying expressions. This means dealing with parentheses/brackets, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right) in the correct order.
• Incorrectly applying the rules of exponents: Make sure you understand and apply the rules of exponents correctly. A common mistake is adding exponents when you should be multiplying them, or vice versa.
• Not simplifying completely: Always simplify your answer as much as possible. This includes expressing numbers as products of their prime factors and eliminating negative exponents.
• Ignoring negative signs: Pay close attention to negative signs, especially when dealing with negative exponents. Remember that a negative exponent indicates a reciprocal.
• Combining terms with different bases: You can only apply the product and quotient of powers rules to terms with the same base. Don't try to combine terms like 2² and 3³ directly.

By being aware of these common mistakes, you can avoid them and simplify exponential expressions with confidence.

Tips and Tricks for Mastering Exponential Expressions

Here are some additional tips and tricks to help you master the art of simplifying exponential expressions:

• Practice regularly: The more you practice, the more comfortable you'll become with the rules of exponents and the process of simplifying expressions. Work through a variety of problems to build your skills.
• Break down complex expressions: If you encounter a complex expression, break it down into smaller, more manageable parts. Simplify each part separately and then combine the results.
• Use prime factorization: Expressing numbers as products of their prime factors is a powerful technique for simplifying exponential expressions. It allows you to apply the rules of exponents more easily.
• Look for patterns: As you practice, you'll start to notice patterns in exponential expressions. Recognizing these patterns can help you simplify expressions more quickly and efficiently.
• Check your work: Always double-check your work to make sure you haven't made any mistakes. It's a good idea to simplify the expression in a different way to verify your answer.

By following these tips and tricks, you can develop a strong understanding of exponential expressions and become proficient at simplifying them.

Conclusion

Simplifying exponential expressions is a fundamental skill in mathematics, and with a solid understanding of the rules of exponents and a systematic approach, it can be mastered by anyone. In this guide, we've walked through the process of simplifying a complex expression step by step, highlighting the key rules and techniques involved. Remember to practice regularly, break down complex expressions, and watch out for common mistakes. With dedication and perseverance, you'll become a pro at simplifying exponential expressions!