Simplifying Exponential Expressions $3^{\frac{12}{5}} \div 3^{-\frac{2}{6}}$

In the realm of mathematics, simplifying expressions is a fundamental skill that allows us to manipulate and understand complex equations with ease. Among the various types of expressions, those involving exponents often present a unique set of challenges. In this comprehensive guide, we will embark on a journey to simplify the expression $3^{\frac{12}{5}} \div 3^{-\frac{2}{6}}$, unraveling the intricacies of dividing powers with the same base and ultimately arriving at the most simplified form.

Understanding the Fundamentals of Exponents

Before we delve into the simplification process, let's first lay a solid foundation by revisiting the fundamental concepts of exponents. An exponent, also known as a power, is a mathematical notation that indicates the number of times a base number is multiplied by itself. For instance, in the expression $3^4$, the base is 3 and the exponent is 4, signifying that 3 is multiplied by itself four times: $3 \times 3 \times 3 \times 3 = 81$.

Exponents play a crucial role in mathematics, allowing us to express large numbers concisely and perform complex calculations with efficiency. They are particularly indispensable when dealing with scientific notation, exponential growth and decay, and various other mathematical concepts.

Dividing Powers with the Same Base: The Quotient Rule

At the heart of simplifying the expression $3^{\frac{12}{5}} \div 3^{-\frac{2}{6}}$ lies the quotient rule of exponents. This rule states that when dividing powers with the same base, we subtract the exponents. Mathematically, this can be expressed as:

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

where 'a' represents the base, and 'm' and 'n' represent the exponents.

The quotient rule is a direct consequence of the definition of exponents. When we divide powers with the same base, we are essentially canceling out common factors. For example, consider the expression $\frac{3^5}{3^2}$. Expanding this, we get:

$\frac{3^5}{3^2} = \frac{3 \times 3 \times 3 \times 3 \times 3}{3 \times 3}$

Canceling out the common factors of 3, we are left with $3 \times 3 \times 3$, which is equal to $3^3$. This illustrates the quotient rule in action: subtracting the exponents (5 - 2) gives us the new exponent of 3.

Applying the Quotient Rule to Our Expression

Now that we have a firm grasp of the quotient rule, let's apply it to simplify the expression $3^{\frac{12}{5}} \div 3^{-\frac{2}{6}}$. The first step is to rewrite the division as a fraction:

$3^{\frac{12}{5}} \div 3^{-\frac{2}{6}} = \frac{3^{\frac{12}{5}}}{3^{-\frac{2}{6}}}$

Next, we can apply the quotient rule by subtracting the exponents:

$\frac{3^{\frac{12}{5}}}{3^{-\frac{2}{6}}} = 3^{\frac{12}{5} - (-\frac{2}{6})}$

Simplifying the exponent subtraction, we get:

$3^{\frac{12}{5} - (-\frac{2}{6})} = 3^{\frac{12}{5} + \frac{2}{6}}$

To add the fractions in the exponent, we need to find a common denominator. The least common multiple of 5 and 6 is 30. So, we rewrite the fractions with a denominator of 30:

$3^{\frac{12}{5} + \frac{2}{6}} = 3^{\frac{12 \times 6}{5 \times 6} + \frac{2 \times 5}{6 \times 5}} = 3^{\frac{72}{30} + \frac{10}{30}}$

Adding the fractions, we get:

$3^{\frac{72}{30} + \frac{10}{30}} = 3^{\frac{82}{30}}$

Simplifying the fraction in the exponent by dividing both numerator and denominator by their greatest common divisor, which is 2, we get:

$3^{\frac{82}{30}} = 3^{\frac{41}{15}}$

Converting to Radical Form (Optional)

While $3^{\frac{41}{15}}$ is a simplified form of the expression, we can further express it in radical form if desired. Recall that a fractional exponent can be interpreted as a radical, where the denominator of the fraction is the index of the radical and the numerator is the power to which the base is raised. In general:

$a^{\frac{m}{n}} = \sqrt[n]{a^m}$

Applying this to our expression, we get:

$3^{\frac{41}{15}} = \sqrt[15]{3^{41}}$

This is the radical form of the simplified expression.

Evaluating the Simplified Expression

To obtain a numerical value for the simplified expression, we can use a calculator. Evaluating $3^{\frac{41}{15}}$, we get approximately 81.

Therefore, the simplified form of the expression $3^{\frac{12}{5}} \div 3^{-\frac{2}{6}}$ is 81.

Conclusion: Mastering Exponential Expressions

Simplifying exponential expressions is a crucial skill in mathematics, and the quotient rule is a powerful tool for dividing powers with the same base. By understanding the fundamentals of exponents and applying the quotient rule systematically, we can effectively simplify complex expressions and gain deeper insights into mathematical relationships. In this guide, we have successfully simplified the expression $3^{\frac{12}{5}} \div 3^{-\frac{2}{6}}$, demonstrating the power and elegance of mathematical principles. Remember, practice is key to mastering any mathematical skill, so continue to explore and challenge yourself with various exponential expressions to solidify your understanding and enhance your problem-solving abilities.

Therefore, the correct answer is D. 81