Simplifying (m^4 N)^-5 / (m^-15 N^6) Using Exponent Rules

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In mathematics, simplifying expressions is a fundamental skill, particularly when dealing with exponents. Exponent rules, also known as laws of exponents, provide a systematic way to manipulate expressions involving powers. These rules streamline the process of simplifying complex expressions, making them easier to understand and work with. This article delves into the application of exponent rules to simplify a given expression, assuming nonzero real numbers. We will focus on the expression (m^4 n)^-5 / (m^-15 n^6), breaking down each step to provide a clear understanding of the process. Mastery of these rules is crucial for success in algebra and beyond, as they form the basis for many advanced mathematical concepts. Understanding the rules for manipulating exponents not only simplifies calculations but also enhances your ability to solve equations and analyze functions. This article aims to provide a comprehensive guide to using exponent rules, ensuring you can confidently simplify any expression involving powers.

Before diving into the simplification process, it's essential to understand the core exponent rules. These rules form the foundation for manipulating expressions with powers. Key exponent rules include:

  1. Product of Powers Rule: *a^m * a^n = a^(m+n). When multiplying powers with the same base, add the exponents.
  2. Quotient of Powers Rule: *a^m / a^n = a^(m-n). When dividing powers with the same base, subtract the exponents.
  3. Power of a Power Rule: (am)n = a^(mn). When raising a power to another power, multiply the exponents.
  4. Power of a Product Rule: *(ab)^n = a^n * b^n. When raising a product to a power, distribute the power to each factor.
  5. Power of a Quotient Rule: *(a/b)^n = a^n / b^n. When raising a quotient to a power, distribute the power to both the numerator and the denominator.
  6. Negative Exponent Rule: *a^-n = 1/a^n. A negative exponent indicates the reciprocal of the base raised to the positive exponent.
  7. Zero Exponent Rule: *a^0 = 1 (where a ≠ 0). Any nonzero number raised to the power of zero is 1.

These rules, when applied correctly, allow for the simplification of complex expressions into more manageable forms. Each rule serves a specific purpose, and recognizing when to apply each one is key to efficient simplification. In the following sections, we will apply these rules step-by-step to simplify the expression (m^4 n)^-5 / (m^-15 n^6).

Now, let's simplify the expression (m^4 n)^-5 / (m^-15 n^6) using the exponent rules. We will break down the process into manageable steps, explaining the application of each rule along the way.

Step 1: Apply the Power of a Product Rule

The first step involves dealing with the term (m^4 n)^-5. According to the Power of a Product Rule, we distribute the exponent -5 to both m^4 and n. This gives us:

(m4)-5 * n^-5

Step 2: Apply the Power of a Power Rule

Next, we simplify (m4)-5 using the Power of a Power Rule, which states that (am)n = a^(mn)*. Multiplying the exponents, we get:

m^(4 * -5) = m^-20

So, the expression becomes:

m^-20 * n^-5

Step 3: Rewrite the Expression

Now, we substitute this back into the original expression:

(m^-20 * n^-5) / (m^-15 * n^6)

Step 4: Apply the Quotient of Powers Rule

To simplify the expression further, we use the Quotient of Powers Rule, which states that a^m / a^n = a^(m-n). We apply this rule separately to the m terms and the n terms:

m^-20 / m^-15 = m^(-20 - (-15)) = m^-5

n^-5 / n^6 = n^(-5 - 6) = n^-11

So, the expression simplifies to:

m^-5 * n^-11

Step 5: Apply the Negative Exponent Rule

Finally, we apply the Negative Exponent Rule to eliminate the negative exponents. This rule states that a^-n = 1/a^n. Applying this to both m^-5 and n^-11, we get:

m^-5 = 1/m^5

n^-11 = 1/n^11

Therefore, the simplified expression is:

(1/m^5) * (1/n^11)

Step 6: Combine Terms

Combine the terms to get the final simplified form:

1 / (m^5 * n^11)

Thus, the simplified form of the expression (m^4 n)^-5 / (m^-15 n^6) is 1 / (m^5 n^11). Each step involved the application of a specific exponent rule, demonstrating how these rules work together to simplify complex expressions.

When simplifying expressions with exponents, several common mistakes can occur. Recognizing these pitfalls can help in avoiding errors and ensuring accurate simplification. Here are some common mistakes to watch out for:

  1. Incorrectly Applying the Power of a Product/Quotient Rule: A frequent error is failing to distribute the exponent correctly over all factors in a product or quotient. For example, (ab)^n should be a^n * b^n, not a^n * b. Similarly, (a/b)^n should be a^n / b^n. Always ensure that the exponent is applied to every term within the parentheses.
  2. Misunderstanding the Negative Exponent Rule: Negative exponents often cause confusion. Remember that a^-n is equal to 1/a^n, not -a^n. The negative exponent indicates a reciprocal, not a negative number. For instance, 2^-3 is 1/2^3 = 1/8, not -8.
  3. Forgetting the Zero Exponent Rule: Any nonzero number raised to the power of zero is 1. This rule is simple but essential. Forgetting this can lead to incorrect simplifications. For example, x^0 is always 1, provided that x is not zero.
  4. Errors in Adding or Subtracting Exponents: When using the Product or Quotient of Powers Rule, ensure the exponents are added or subtracted correctly. Double-check your arithmetic to avoid mistakes. For example, a^5 / a^2 should be a^(5-2) = a^3, not a^7 or a^10.
  5. Ignoring the Order of Operations: Exponentiation should be performed before multiplication, division, addition, and subtraction. Following the correct order of operations (PEMDAS/BODMAS) is crucial for accurate simplification. For example, in the expression 2 * 3^2, you should first calculate 3^2 = 9 and then multiply by 2, resulting in 18, not (2 * 3)^2 = 36.

By being mindful of these common errors and practicing the exponent rules diligently, you can improve your accuracy and confidence in simplifying expressions.

To solidify your understanding of exponent rules, working through practice problems is essential. Here are a few problems to help you practice simplifying expressions. Try to apply the rules we've discussed, and remember to avoid common mistakes.

  1. Simplify: (x^3 y-2)4 / (x^-2 y^3)
  2. Simplify: ((a^2 b)^-3 * c^4) / (a^-5 b^2 c^-1)
  3. Simplify: (4p^0 q^-2 r3)-2
  4. Simplify: ((2m^-3 n2)2) / (8m^4 n^-3)
  5. Simplify: ((5x^2 y^-1 z3)-1) / (10x^-3 y^2 z^-2)

Solutions:

  1. x^14 / y^11
  2. a^-1 b^-5 c^5
  3. q^4 / (16r^6)
  4. n^7 / (2m^10)
  5. x / (50y z)

Working through these problems will help you become more comfortable with the exponent rules and improve your ability to simplify complex expressions. Practice is key to mastering these concepts, so don't hesitate to try additional problems and seek help when needed.

In conclusion, simplifying expressions using exponent rules is a fundamental skill in mathematics. By understanding and applying these rules correctly, complex expressions can be transformed into simpler, more manageable forms. We have explored the key exponent rules, including the Product of Powers Rule, Quotient of Powers Rule, Power of a Power Rule, Power of a Product Rule, Power of a Quotient Rule, Negative Exponent Rule, and Zero Exponent Rule. We also walked through a step-by-step simplification of the expression (m^4 n)^-5 / (m^-15 n^6), illustrating the practical application of these rules.

Additionally, we highlighted common mistakes to avoid, such as incorrectly applying the Power of a Product/Quotient Rule, misunderstanding the Negative Exponent Rule, forgetting the Zero Exponent Rule, errors in adding or subtracting exponents, and ignoring the order of operations. Being aware of these pitfalls will help you maintain accuracy in your simplifications.

Finally, we provided practice problems with solutions to reinforce your understanding. Consistent practice is crucial for mastering exponent rules and building confidence in simplifying expressions. These skills are not only essential for algebra but also for more advanced mathematical concepts. By dedicating time to understanding and practicing exponent rules, you will strengthen your mathematical foundation and enhance your problem-solving abilities.