Rewriting & Evaluating Expressions With Exponent Properties

by Alex Johnson 60 views

Let's dive into the world of exponents and learn how to simplify and evaluate expressions. In this article, we'll break down the expression (11j⁻³k⁻²)(j³k⁴) using the properties of exponents. We will then evaluate the simplified expression when j = -8 and k = 7. Understanding these concepts is crucial for anyone studying algebra and beyond. So, grab a pen and paper, and let's get started!

Understanding the Properties of Exponents

Before we jump into the specific problem, let's take a moment to review the fundamental properties of exponents. These rules are the foundation for simplifying expressions, and mastering them will make your life much easier. Here are the key properties we'll be using:

  • Product of Powers: When multiplying exponents with the same base, you add the powers: aᵐ * aⁿ = aᵐ⁺ⁿ
  • Quotient of Powers: When dividing exponents with the same base, you subtract the powers: aᵐ / aⁿ = aᵐ⁻ⁿ
  • Power of a Power: When raising a power to another power, you multiply the exponents: (aᵐ)ⁿ = aᵐⁿ
  • Power of a Product: When raising a product to a power, you distribute the power to each factor: (ab)ⁿ = aⁿbⁿ
  • Power of a Quotient: When raising a quotient to a power, you distribute the power to both the numerator and the denominator: (a/b)ⁿ = aⁿ/bⁿ
  • Negative Exponent: A negative exponent indicates a reciprocal: a⁻ⁿ = 1/aⁿ
  • Zero Exponent: Any non-zero number raised to the power of zero is 1: a⁰ = 1

These properties might seem like a lot to remember, but with practice, they'll become second nature. Keep them handy as we work through our example problem.

Rewriting the Expression: (11j⁻³k⁻²)(j³k⁴)

Now, let's tackle the expression (11j⁻³k⁻²)(j³k⁴). Our goal here is to simplify it using the properties of exponents we just discussed. The most important rule we'll use here is the Product of Powers rule. We'll combine the terms with the same base by adding their exponents.

  1. Identify Like Terms: First, we need to identify the terms with the same base. In this expression, we have terms with the bases j and k. The constant 11 will remain separate for now.

  2. Apply the Product of Powers Rule:

    • For the j terms: j⁻³ * j³ = j⁻³⁺³ = j⁰
    • For the k terms: k⁻² * k⁴ = k⁻²⁺⁴ = k²

  3. Rewrite the Expression: Now, we can rewrite the expression using our simplified terms:

    • (11j⁻³k⁻²)(j³k⁴) = 11 * j⁰ * k²

  4. Apply the Zero Exponent Rule: Remember that any non-zero number raised to the power of zero is 1. So, j⁰ = 1.

  5. Final Simplified Expression: Substituting j⁰ with 1, we get our final simplified expression:

    • 11 * 1 * k² = 11k²

So, we've successfully rewritten the expression (11j⁻³k⁻²)(j³k⁴) as 11k². This simplified form is much easier to work with when we need to evaluate it for specific values of j and k.

Evaluating the Rewritten Expression When j = -8 and k = 7

Now that we have our simplified expression, 11k², it's time to evaluate it when j = -8 and k = 7. Notice that the variable j has disappeared from our simplified expression. This means the value of j won't affect the final result. We only need to substitute the value of k into the expression.

  1. Substitute the Value of k: Replace k with 7 in the expression 11k²:

    • 11k² = 11(7)²

  2. Calculate the Exponent: First, we need to calculate 7 squared, which is 7 * 7 = 49.

    • 11(7)² = 11 * 49

  3. Multiply: Now, multiply 11 by 49:

    • 11 * 49 = 539

Therefore, when j = -8 and k = 7, the value of the expression (11j⁻³k⁻²)(j³k⁴) is 539. We arrived at this answer by first simplifying the expression using the properties of exponents and then substituting the given values. This process highlights the power of simplification in mathematics.

Common Mistakes to Avoid

Working with exponents can be tricky, and it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

  • Incorrectly Applying the Product of Powers Rule: Remember, you only add exponents when multiplying terms with the same base. Don't try to add exponents of terms with different bases.
  • Forgetting the Order of Operations: Always follow the order of operations (PEMDAS/BODMAS). Exponents should be calculated before multiplication and division.
  • Misunderstanding Negative Exponents: A negative exponent indicates a reciprocal, not a negative number. a⁻ⁿ is equal to 1/aⁿ, not -aⁿ.
  • Ignoring the Zero Exponent Rule: Any non-zero number raised to the power of zero is 1. Don't forget this rule!
  • Distributing Exponents Incorrectly: When raising a product or quotient to a power, remember to distribute the exponent to every factor or term. For example, (ab)ⁿ = aⁿbⁿ, not abⁿ.

By being aware of these common mistakes, you can avoid them and improve your accuracy when working with exponents. Practice makes perfect, so keep working on these types of problems!

Practice Problems

To solidify your understanding of exponent properties, try working through these practice problems:

  1. Simplify and evaluate: (3x²y⁻¹)(2x⁻¹y³) when x = 4 and y = -2
  2. Simplify: (5a³b²)² / (10a⁻¹b⁴)
  3. Rewrite with positive exponents: (4m⁻⁵n²)⁻¹

Working through these problems will help you gain confidence in applying the properties of exponents. Don't be afraid to make mistakes – they're a valuable part of the learning process. If you get stuck, review the properties and examples we discussed earlier.

Conclusion

In this article, we've explored how to rewrite and evaluate expressions using the properties of exponents. We successfully simplified the expression (11j⁻³k⁻²)(j³k⁴) to 11k² and then evaluated it to be 539 when j = -8 and k = 7. We also discussed common mistakes to avoid and provided practice problems to help you master these concepts. Understanding exponent properties is a fundamental skill in algebra and will serve you well in more advanced mathematics.

Keep practicing, and you'll become a pro at working with exponents! For further exploration and practice, you can visit resources like Khan Academy's Exponent Rules. Happy calculating!