Simplifying Expressions Dividing Monomials With Positive Exponents

When dividing monomials, it's crucial to understand the rules of exponents and how they apply to both coefficients and variables. Let's dive into simplifying the expression $\frac{-56x^9}{8x^{15}}$ and ensure our answer is written with positive exponents. This process involves dividing the coefficients and applying the quotient rule of exponents, which states that when dividing like bases, you subtract the exponents.

Step-by-Step Solution

1. Divide the Coefficients: Begin by dividing the numerical coefficients. In this case, we have -56 divided by 8, which equals -7. This is a straightforward arithmetic operation, but it's essential to maintain the negative sign.

2. Apply the Quotient Rule of Exponents: The quotient rule states that $\frac{a^m}{a^n} = a^{m-n}$. For our variables, we have $\frac{x^9}{x^{15}}$. Applying the rule, we subtract the exponents: 9 - 15 = -6. This gives us $x^{-6}$.

3. Combine the Results: Now, we combine the results from the coefficient division and the exponent subtraction. We have -7 and $x^{-6}$, so our expression becomes $-7x^{-6}$.

4. Eliminate Negative Exponents: The final step is to ensure we have only positive exponents in our answer. A negative exponent indicates a reciprocal. Specifically, $x^{-6}$ is the same as $\frac{1}{x^6}$. Therefore, we rewrite $-7x^{-6}$ as $-7 \cdot \frac{1}{x^6}$.

5. Final Simplified Expression: Combining the terms, we get our final simplified expression: $\frac{-7}{x^6}$. This expression has a negative coefficient but a positive exponent, fulfilling the requirement of the problem.

Understanding the Quotient Rule of Exponents

The quotient rule of exponents is a fundamental concept in algebra. It allows us to simplify expressions involving the division of powers with the same base. The rule is mathematically expressed as:

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

Where a is the base, and m and n are the exponents. This rule is derived from the basic principles of exponents, which define exponents as a shorthand for repeated multiplication. When you divide two exponential terms with the same base, you are essentially canceling out common factors. Subtracting the exponents is a way to account for the remaining factors.

To illustrate, let's consider a simple example:

$$\frac{x^5}{x^2}$$

Here, $x^5$ means $x \cdot x \cdot x \cdot x \cdot x$, and $x^2$ means $x \cdot x$. When we divide $x^5$ by $x^2$, we get:

$$\frac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x}$$

We can cancel out two $x$ terms from both the numerator and the denominator, leaving us with:

$$x \cdot x \cdot x = x^3$$

Applying the quotient rule directly, we subtract the exponents: 5 - 2 = 3, which gives us $x^3$. This confirms the rule.

Common Mistakes and How to Avoid Them

When working with the quotient rule, several common mistakes can occur. Understanding these pitfalls can help in avoiding them:

1. Forgetting to Subtract Exponents: One of the most common errors is simply forgetting to subtract the exponents when dividing like bases. Always remember that division of exponential terms requires subtracting the exponents.

2. Incorrectly Applying the Rule to Coefficients: The quotient rule applies to exponents, not coefficients. Coefficients should be divided normally. For instance, in the expression $\frac{4x^5}{2x^2}$, divide 4 by 2 to get 2, and then apply the quotient rule to the exponents.

3. Dealing with Negative Exponents: Negative exponents can be confusing. Remember that a negative exponent means taking the reciprocal of the base raised to the positive exponent. For example, $x^{-3} = \frac{1}{x^3}$. Always rewrite negative exponents as positive exponents in the final answer.

4. Mixing Up Quotient and Product Rules: It’s easy to mix up the quotient rule (subtraction of exponents) with the product rule (addition of exponents). Remember, when you multiply like bases, you add the exponents ($a^m \cdot a^n = a^{m+n}$), and when you divide, you subtract them.

5. Not Simplifying Completely: Always ensure that the expression is fully simplified. This means all coefficients are reduced, and all negative exponents have been converted to positive exponents.

Examples to Illustrate Common Mistakes

• Mistake 1: Not subtracting exponents

  • Incorrect: $\frac{x^7}{x^3} = x^{7 \cdot 3} = x^{21}$
  • Correct: $\frac{x^7}{x^3} = x^{7-3} = x^4$

• Mistake 2: Incorrectly applying the rule to coefficients

  • Incorrect: $\frac{6x^4}{2x^2} = 3x^{4-2} = x^2$
  • Correct: $\frac{6x^4}{2x^2} = 3x^{4-2} = 3x^2$

• Mistake 3: Misunderstanding negative exponents

  • Incorrect: $4x^{-2} = \frac{1}{4x^2}$
  • Correct: $4x^{-2} = 4 \cdot \frac{1}{x^2} = \frac{4}{x^2}$

By understanding these common mistakes and practicing the correct application of the quotient rule, you can confidently simplify complex exponential expressions.

Working with Negative Exponents

Understanding negative exponents is vital for simplifying expressions in algebra. A negative exponent indicates the reciprocal of the base raised to the positive exponent. In other words:

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

Where a is the base and n is the exponent. This concept is crucial for rewriting expressions in their simplest form and for ensuring that all exponents are positive, as often required in mathematical problems.

The reason behind this rule lies in the patterns of exponents. Consider the sequence of powers of 2:

$$2^3 = 8$$

$$2^2 = 4$$

$$2^1 = 2$$

$$2^0 = 1$$

Each time the exponent decreases by 1, the value is divided by 2. Following this pattern, if we continue to decrease the exponent:

$$2^{-1} = \frac{1}{2}$$

$$2^{-2} = \frac{1}{4}$$

$$2^{-3} = \frac{1}{8}$$

This pattern illustrates that a negative exponent represents the reciprocal of the positive exponent. Therefore, $2^{-n}$ is the same as $\frac{1}{2^n}$.

Rules for Manipulating Negative Exponents

To effectively work with negative exponents, it’s essential to understand the rules that govern their manipulation:

1. Converting Negative to Positive Exponents: The most fundamental rule is to rewrite a term with a negative exponent as its reciprocal with a positive exponent. For example:

   • $x^{-4} = \frac{1}{x^4}$
   • $5^{-2} = \frac{1}{5^2} = \frac{1}{25}$

2. Moving Terms Between Numerator and Denominator: A term with a negative exponent in the numerator can be moved to the denominator with a positive exponent, and vice versa. For example:

   • $\frac{1}{x^{-3}} = x^3$
   • $\frac{y^{-2}}{z^{-5}} = \frac{z^5}{y^2}$

3. Applying the Product Rule with Negative Exponents: When multiplying like bases with negative exponents, add the exponents, just as with positive exponents. For example:

   • $x^{-2} \cdot x^5 = x^{-2+5} = x^3$
   • $a^{-3} \cdot a^{-1} = a^{-3+(-1)} = a^{-4} = \frac{1}{a^4}$

4. Applying the Quotient Rule with Negative Exponents: When dividing like bases with negative exponents, subtract the exponents. For example:

   • $\frac{x^{-4}}{x^{-2}} = x^{-4-(-2)} = x^{-4+2} = x^{-2} = \frac{1}{x^2}$
   • $\frac{y^3}{y^{-1}} = y^{3-(-1)} = y^{3+1} = y^4$

5. Power of a Power Rule with Negative Exponents: When raising a power to a power, multiply the exponents. For example:

   • $(x^{-2})^3 = x^{-2 \cdot 3} = x^{-6} = \frac{1}{x^6}$
   • $(a^{-1})^{-4} = a^{-1 \cdot -4} = a^4$

Practical Examples of Simplifying Expressions with Negative Exponents

To solidify understanding, let's work through several examples:

Example 1: Simplify $3x^{-2}y^4 \cdot 5x^3y^{-1}$

1. Multiply the coefficients: $3 \cdot 5 = 15$
2. Apply the product rule to the $x$ terms: $x^{-2} \cdot x^3 = x^{-2+3} = x^1 = x$
3. Apply the product rule to the $y$ terms: $y^4 \cdot y^{-1} = y^{4+(-1)} = y^3$
4. Combine the results: $15xy^3$

Example 2: Simplify $\frac{12a^{-3}b^5}{4a^2b^{-2}}$

1. Divide the coefficients: $\frac{12}{4} = 3$
2. Apply the quotient rule to the $a$ terms: $\frac{a^{-3}}{a^2} = a^{-3-2} = a^{-5} = \frac{1}{a^5}$
3. Apply the quotient rule to the $b$ terms: $\frac{b^5}{b^{-2}} = b^{5-(-2)} = b^{5+2} = b^7$
4. Combine the results: $3 \cdot \frac{1}{a^5} \cdot b^7 = \frac{3b^7}{a^5}$

Example 3: Simplify $(\frac{2x^{-1}y^3}{z^{-2}})^{-2}$

1. Apply the power of a power rule to each term inside the parentheses:
   • $2^{-2} = \frac{1}{2^2} = \frac{1}{4}$
   • $(x^{-1})^{-2} = x^{-1 \cdot -2} = x^2$
   • $(y^3)^{-2} = y^{3 \cdot -2} = y^{-6} = \frac{1}{y^6}$
   • $(z^{-2})^{-2} = z^{-2 \cdot -2} = z^4$
2. Combine the results: $\frac{1}{4} \cdot x^2 \cdot \frac{1}{y^6} \cdot z^4 = \frac{x^2z^4}{4y^6}$

Understanding and practicing these rules and examples will significantly enhance your ability to simplify expressions involving negative exponents.

Conclusion

In summary, simplifying the expression $\frac{-56x^9}{8x^{15}}$ involves dividing coefficients, applying the quotient rule of exponents, and ensuring all exponents are positive. The final simplified expression is $\frac{-7}{x^6}$. This process highlights the importance of understanding and applying the rules of exponents correctly. By following these steps, you can confidently simplify similar expressions and master the art of dividing monomials.