Simplify: (-6a⁻⁴b⁷)/(-2a⁻¹b⁻⁷) - A Step-by-Step Guide

Algebraic expressions can sometimes look intimidating, but with a systematic approach, they can be simplified to reveal their underlying simplicity. In this article, we'll walk through the process of simplifying the expression (-6a⁻⁴b⁷)/(-2a⁻¹b⁻⁷) step by step. This guide aims to provide a clear and comprehensive understanding of the rules and techniques involved in simplifying such expressions. Let's dive in!

Understanding the Basics

Before we begin, let's revisit some fundamental concepts in algebra that will be useful for this simplification.

• Negative Exponents: A term with a negative exponent can be rewritten as its reciprocal with a positive exponent. For example, a⁻ⁿ = 1/aⁿ.
• Division of Like Bases: When dividing terms with the same base, subtract the exponents. For example, aᵐ / aⁿ = aᵐ⁻ⁿ.
• Simplifying Fractions: Numerical fractions should be simplified by dividing both the numerator and the denominator by their greatest common divisor.

With these basics in mind, we can proceed to simplify the given expression.

Step-by-Step Simplification

Let's break down the expression (-6a⁻⁴b⁷)/(-2a⁻¹b⁻⁷) into manageable parts and simplify each one.

1. Simplify the Numerical Coefficients

The numerical coefficients in the expression are -6 and -2. We can simplify the fraction -6/-2 as follows:

-6 / -2 = 3

So, the simplified numerical coefficient is 3.

2. Simplify the 'a' Terms

We have a⁻⁴ in the numerator and a⁻¹ in the denominator. Using the rule for dividing like bases, we subtract the exponents:

a⁻⁴ / a⁻¹ = a⁻⁴⁻⁽⁻¹⁾ = a⁻⁴⁺¹ = a⁻³

So, the simplified 'a' term is a⁻³.

3. Simplify the 'b' Terms

We have b⁷ in the numerator and b⁻⁷ in the denominator. Again, using the rule for dividing like bases, we subtract the exponents:

b⁷ / b⁻⁷ = b⁷⁻⁽⁻⁷⁾ = b⁷⁺⁷ = b¹⁴

So, the simplified 'b' term is b¹⁴.

4. Combine the Simplified Terms

Now that we have simplified the numerical coefficients, the 'a' terms, and the 'b' terms, we can combine them to get the simplified expression:

3 * a⁻³ * b¹⁴ = 3a⁻³b¹⁴

5. Eliminate the Negative Exponent

To eliminate the negative exponent in a⁻³, we can rewrite it as its reciprocal with a positive exponent:

a⁻³ = 1/a³

So, the expression becomes:

3 * (1/a³) * b¹⁴ = (3b¹⁴)/a³

Therefore, the simplified expression is (3b¹⁴)/a³.

Detailed Explanation

To ensure a complete understanding, let's delve deeper into each step with more detailed explanations and examples.

Simplifying Numerical Coefficients in Detail

The numerical coefficients are the constant numbers that multiply the variable terms. In our expression (-6a⁻⁴b⁷)/(-2a⁻¹b⁻⁷), the numerical coefficients are -6 and -2. Simplifying these involves dividing -6 by -2. When dividing two negative numbers, the result is positive. Therefore, -6 divided by -2 equals 3. This step is straightforward but crucial, as it sets the base for the rest of the simplification.

Example:

Consider the expression (12x²)/(4y). The numerical coefficients are 12 and 4. Simplifying 12/4 gives us 3. Thus, the expression simplifies to 3(x²/y).

Simplifying 'a' Terms in Detail

Simplifying the 'a' terms involves using the quotient rule for exponents, which states that when you divide like bases, you subtract the exponents. In our case, we have a⁻⁴ divided by a⁻¹. The rule is aᵐ / aⁿ = aᵐ⁻ⁿ. Thus, a⁻⁴ / a⁻¹ = a⁻⁴⁻⁽⁻¹⁾. This simplifies to a⁻⁴⁺¹, which equals a⁻³. A negative exponent means that we take the reciprocal of the base raised to the positive exponent. Therefore, a⁻³ is the same as 1/a³.

Example:

Consider the expression (x⁵)/(x²). Using the quotient rule, we get x⁵⁻² = x³. Similarly, if we have (x⁻²)/(x⁻⁵), we get x⁻²⁻⁽⁻⁵⁾ = x⁻²⁺⁵ = x³.

Simplifying 'b' Terms in Detail

Similarly to the 'a' terms, we apply the quotient rule for exponents to the 'b' terms. We have b⁷ divided by b⁻⁷. Thus, b⁷ / b⁻⁷ = b⁷⁻⁽⁻⁷⁾. This simplifies to b⁷⁺⁷, which equals b¹⁴. The 'b' term is now fully simplified as it has a positive exponent.

Example:

Consider the expression (y⁻³)/(y⁻⁸). Using the quotient rule, we get y⁻³⁻⁽⁻⁸⁾ = y⁻³⁺⁸ = y⁵. If we have (y¹⁰)/(y⁵), we get y¹⁰⁻⁵ = y⁵.

Combining and Final Simplification in Detail

After simplifying the numerical coefficients and the 'a' and 'b' terms, we combine them to form the simplified expression. We found that the numerical coefficients simplify to 3, the 'a' terms simplify to a⁻³, and the 'b' terms simplify to b¹⁴. Combining these, we get 3 * a⁻³ * b¹⁴ = 3a⁻³b¹⁴. To eliminate the negative exponent, we rewrite a⁻³ as 1/a³. Thus, the final simplified expression is (3b¹⁴)/a³.

Example:

Consider simplifying (8x⁻⁵y²)/(2x⁻²y⁻¹). First, simplify the numerical coefficients: 8/2 = 4. Then, simplify the 'x' terms: x⁻⁵ / x⁻² = x⁻⁵⁻⁽⁻²⁾ = x⁻³. Simplify the 'y' terms: y² / y⁻¹ = y²⁻⁽⁻¹⁾ = y³. Combining these, we get 4x⁻³y³. Eliminating the negative exponent, we get (4y³)/x³.

Common Mistakes to Avoid

When simplifying algebraic expressions, there are several common mistakes that students often make. Being aware of these mistakes can help you avoid them and ensure accurate simplification.

1. Incorrectly Applying the Quotient Rule

A common mistake is to add the exponents instead of subtracting them when dividing like bases. Remember that aᵐ / aⁿ = aᵐ⁻ⁿ, not aᵐ⁺ⁿ. Always subtract the exponent in the denominator from the exponent in the numerator.

2. Misunderstanding Negative Exponents

Negative exponents indicate reciprocals, not negative numbers. For example, a⁻ⁿ = 1/aⁿ, not -aⁿ. Be careful to rewrite terms with negative exponents correctly before proceeding with further simplification.

3. Forgetting to Distribute Negative Signs

When subtracting exponents, especially when dealing with negative exponents, it's crucial to distribute the negative sign correctly. For instance, a⁻² / a⁻⁵ = a⁻²⁻⁽⁻⁵⁾ = a⁻²⁺⁵ = a³. Failing to distribute the negative sign can lead to incorrect results.

4. Not Simplifying Numerical Coefficients

Always simplify numerical coefficients before dealing with variable terms. This can make the expression easier to manage and reduce the chances of making mistakes later on.

5. Incorrectly Combining Terms

Only terms with the same base can have their exponents combined when dividing. For example, you cannot combine the exponents of 'a' and 'b' terms. Keep the terms separate and simplify them individually.

Practice Problems

To solidify your understanding, here are a few practice problems. Try to simplify them using the steps outlined in this article. Solutions are provided below.

Practice Problem 1

Simplify: (15x⁻²y⁵) / (3x³y⁻²)

Practice Problem 2

Simplify: (-8a⁴b⁻³) / (4a⁻¹b²)

Practice Problem 3

Simplify: (20p⁻⁶q⁻⁴) / (5p⁻²q⁵)

Solutions to Practice Problems

Solution to Practice Problem 1

(15x⁻²y⁵) / (3x³y⁻²) = 5x⁻⁵y⁷ = (5y⁷) / x⁵

Solution to Practice Problem 2

(-8a⁴b⁻³) / (4a⁻¹b²) = -2a⁵b⁻⁵ = (-2a⁵) / b⁵

Solution to Practice Problem 3

(20p⁻⁶q⁻⁴) / (5p⁻²q⁵) = 4p⁻⁴q⁻⁹ = 4 / (p⁴q⁹)

Conclusion

Simplifying algebraic expressions like (-6a⁻⁴b⁷)/(-2a⁻¹b⁻⁷) might seem challenging at first, but by breaking down the problem into smaller, manageable steps, it becomes much easier. Remember to simplify numerical coefficients, apply the quotient rule for exponents, handle negative exponents correctly, and combine the simplified terms. With practice and attention to detail, you can master the art of simplifying algebraic expressions. Regularly practicing similar problems will boost your confidence and accuracy.

For more information on algebraic expressions and simplification techniques, visit Khan Academy's Algebra Section.