Simplifying Algebraic Expressions A Detailed Guide To -2/3ab^3 ÷ (-1/6ab)^2

In the realm of algebra, simplifying expressions often involves navigating through a labyrinth of variables, exponents, and coefficients. This article delves into the step-by-step simplification of the algebraic expression $-\frac{2}{3} a b^3 \div\left(-\frac{1}{6} a b\right)^2$, providing a comprehensive guide for students and enthusiasts alike. Mastering such simplifications is crucial for a solid foundation in algebra and its applications in various fields of mathematics and science.

Understanding the Fundamentals of Algebraic Division

At its core, algebraic division is the inverse operation of multiplication. When we divide one algebraic expression by another, we are essentially finding a third expression that, when multiplied by the divisor, yields the dividend. In our case, the dividend is $-\frac{2}{3} a b^3$ and the divisor is $\left(-\frac{1}{6} a b\right)^2$. To effectively tackle this division, it's essential to have a firm grasp on the fundamental rules of exponents, fractions, and the order of operations (PEMDAS/BODMAS).

Exponents play a pivotal role in algebraic expressions, indicating the number of times a base is multiplied by itself. For instance, $b^3$ signifies $b \times b \times b$. Understanding how exponents interact with multiplication and division is crucial. For example, when dividing terms with the same base, we subtract the exponents (e.g., $x^m / x^n = x^{m-n}$). Furthermore, we must remember the power of a product rule: $(xy)^n = x^n y^n$, which is particularly relevant when dealing with expressions raised to a power, as seen in our divisor, $\left(-\frac{1}{6} a b\right)^2$. This rule dictates that we apply the exponent to each factor within the parentheses.

Fractions are another cornerstone of algebraic expressions. A fraction represents a part of a whole and consists of a numerator (the top number) and a denominator (the bottom number). When dividing fractions, we invert the divisor (the fraction we are dividing by) and multiply. This seemingly simple rule is paramount in simplifying our given expression. Additionally, we must be comfortable with simplifying fractions by canceling out common factors between the numerator and the denominator.

The order of operations, often remembered by the acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction), provides a roadmap for simplifying expressions. It dictates the sequence in which operations should be performed to arrive at the correct answer. In our case, we must first address the exponent in the divisor before proceeding with the division.

Step-by-Step Simplification of $-\frac{2}{3} a b^3 \div\left(-\frac{1}{6} a b\right)^2$

Now, let's embark on the journey of simplifying the expression $-\frac{2}{3} a b^3 \div\left(-\frac{1}{6} a b\right)^2$, breaking it down into manageable steps. By meticulously following each step, we'll not only arrive at the simplified form but also gain a deeper understanding of the underlying principles.

Step 1: Simplify the Divisor

Our first task is to simplify the divisor, $\left(-\frac{1}{6} a b\right)^2$. This involves applying the power of a product rule, which states that $(xy)^n = x^n y^n$. We distribute the exponent 2 to each factor within the parentheses:

$\left(-\frac{1}{6} a b\right)^2 = \left(-\frac{1}{6}\right)^2 \cdot a^2 \cdot b^2$

Now, we evaluate each term individually:

• $\left(-\frac{1}{6}\right)^2 = \left(-\frac{1}{6}\right) \times \left(-\frac{1}{6}\right) = \frac{1}{36}$
• $a^2$ remains as $a^2$
• $b^2$ remains as $b^2$

Thus, the simplified divisor is $\frac{1}{36} a^2 b^2$.

Step 2: Rewrite the Division as Multiplication

Division can be transformed into multiplication by inverting the divisor and multiplying. This is a fundamental rule when dealing with fractions and algebraic expressions alike. So, we rewrite the original expression as:

$-\frac{2}{3} a b^3 \div \left(\frac{1}{36} a^2 b^2\right) = -\frac{2}{3} a b^3 \times \frac{36}{1 a^2 b^2}$

Notice how the divisor $\frac{1}{36} a^2 b^2$ has been inverted to $\frac{36}{1 a^2 b^2}$. This transformation is the key to simplifying the expression further.

Step 3: Multiply the Fractions

Now that we have rewritten the division as multiplication, we can multiply the fractions by multiplying the numerators and the denominators separately:

$-\frac{2}{3} a b^3 \times \frac{36}{1 a^2 b^2} = -\frac{2 \times 36}{3 \times 1} \times \frac{a b^3}{a^2 b^2}$

This step combines the coefficients and the variable terms, making it easier to simplify in the subsequent steps.

Step 4: Simplify the Coefficients

Let's simplify the numerical coefficients first. We have:

$-\frac{2 \times 36}{3 \times 1} = -\frac{72}{3}$

Dividing 72 by 3, we get:

$-\frac{72}{3} = -24$

So, the simplified coefficient is -24.

Step 5: Simplify the Variable Terms

Now, let's tackle the variable terms. We have $\frac{a b^3}{a^2 b^2}$. Remember the rule for dividing exponents with the same base: $x^m / x^n = x^{m-n}$. Applying this rule, we get:

• For $a$: $\frac{a}{a^2} = a^{1-2} = a^{-1}$
• For $b$: $\frac{b^3}{b^2} = b^{3-2} = b^1 = b$

Therefore, the simplified variable term is $a^{-1}b$.

Step 6: Combine the Simplified Terms

Finally, we combine the simplified coefficient and the simplified variable terms:

$-24 \times a^{-1} b = -24 a^{-1} b$

It's customary to express algebraic expressions without negative exponents. To eliminate the negative exponent, we use the rule $x^{-n} = \frac{1}{x^n}$. Applying this rule, we get:

$-24 a^{-1} b = -24 \times \frac{1}{a} \times b = -\frac{24b}{a}$

The Final Simplified Expression

After meticulously following each step, we have successfully simplified the expression $-\frac{2}{3} a b^3 \div\left(-\frac{1}{6} a b\right)^2$ to its final form:

$-\frac{24b}{a}$

This simplified expression is more concise and easier to work with in further algebraic manipulations. The journey through the simplification process has not only provided us with the answer but also reinforced our understanding of the fundamental principles of algebraic division, exponents, and fractions.

Common Pitfalls to Avoid

Simplifying algebraic expressions can be tricky, and it's easy to stumble along the way. Here are some common pitfalls to watch out for:

• Forgetting the Order of Operations: Always adhere to PEMDAS/BODMAS. Exponents must be dealt with before multiplication or division.
• Incorrectly Applying the Power of a Product Rule: Remember that the exponent applies to every factor within the parentheses. A common mistake is to apply the exponent only to the variable and not the coefficient.
• Errors in Fraction Division: Dividing by a fraction is the same as multiplying by its inverse. Ensure you invert the divisor and not the dividend.
• Misunderstanding Exponent Rules: The rules for exponents can be confusing. Remember that when dividing terms with the same base, you subtract the exponents, and when multiplying, you add them.
• Ignoring Negative Signs: Negative signs are like landmines in algebraic expressions. Be extra careful when dealing with them, especially when squaring negative numbers.

By being mindful of these common pitfalls, you can significantly reduce the chances of making errors and improve your algebraic simplification skills.

Practice Makes Perfect

Mastering the simplification of algebraic expressions, like any mathematical skill, requires practice. The more you practice, the more comfortable and confident you will become. Here are some suggestions for honing your skills:

• Work Through Numerous Examples: Seek out a variety of examples, ranging from simple to complex, and work through them step by step. Pay attention to the reasoning behind each step and identify any patterns or shortcuts.
• Utilize Online Resources: There are numerous websites and online platforms that offer practice problems and step-by-step solutions. These resources can be invaluable for reinforcing your understanding and identifying areas where you need further practice.
• Seek Help When Needed: Don't hesitate to ask for help from teachers, tutors, or classmates if you're struggling with a particular concept or problem. A fresh perspective can often shed light on a difficult situation.
• Create Your Own Problems: A great way to solidify your understanding is to create your own problems and solve them. This forces you to think critically about the concepts and apply them in new ways.
• Review Regularly: Make it a habit to review previously learned concepts and skills regularly. This will help you retain the information and prevent forgetting.

Algebraic simplification is a fundamental skill that underpins many areas of mathematics and its applications. By understanding the principles, avoiding common pitfalls, and practicing diligently, you can master this skill and unlock a world of mathematical possibilities.

In conclusion, simplifying $-\frac{2}{3} a b^3 \div\left(-\frac{1}{6} a b\right)^2$ to $-\frac{24b}{a}$ exemplifies the power of applying fundamental algebraic principles. This step-by-step guide, along with the insights into common pitfalls and practice strategies, serves as a valuable resource for anyone seeking to enhance their algebraic skills. So, embrace the challenge, practice consistently, and watch your algebraic prowess flourish.