Simplifying Algebraic Expressions A Step-by-Step Guide To $18 X^2 \sqrt{14 X^8} \div 6 \sqrt{7 X^4}$

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In the realm of mathematics, algebraic expressions often present themselves as intricate puzzles, challenging us to unravel their complexities and reveal their underlying simplicity. One such puzzle involves simplifying expressions with radicals and exponents. Let's embark on a journey to dissect the expression $18 x^2 \sqrt{14 x^8} \div 6 \sqrt{7 x^4}$, where $x \neq 0$, and discover its equivalent form. This expression, at first glance, might seem daunting, but with a systematic approach and a firm grasp of mathematical principles, we can navigate through its intricacies and arrive at the solution. The key lies in understanding the properties of radicals, exponents, and division, and how they interact with each other. By carefully applying these principles, we can transform the expression into a more manageable form, paving the way for simplification.

Deconstructing the Expression The initial step in simplifying any mathematical expression is to break it down into its fundamental components. In our case, we have an expression involving radicals, exponents, and division. Let's dissect each part to gain a clearer understanding. We have the term $18x^2$, which is a product of a constant and a variable raised to a power. Then, we encounter the radical expression $\sqrt{14x^8}$, which involves a square root and a variable raised to a power within the radical. Next, we have the division operation, indicated by the symbol $\div$, and finally, we have another radical expression $6\sqrt{7x^4}$. By identifying these components, we can begin to strategize how to simplify the expression. The presence of radicals and exponents suggests that we will need to apply the properties of these operations to manipulate the expression. The division operation indicates that we will need to consider how to simplify fractions involving radicals and exponents. With a clear understanding of the components, we can now move on to the next step: applying mathematical principles to simplify the expression.

Applying the Quotient Rule for Radicals

The quotient rule for radicals states that the square root of a quotient is equal to the quotient of the square roots. Mathematically, this can be expressed as $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$, where $a$ and $b$ are non-negative numbers and $b \neq 0$. This rule provides us with a powerful tool for simplifying expressions involving radicals and division. In our case, we can apply this rule to combine the two radical terms in the expression. By applying the quotient rule, we can rewrite the expression as a single radical term, making it easier to manipulate and simplify. This step is crucial in our journey to unravel the expression's complexity and reveal its equivalent form. The quotient rule allows us to consolidate the radical terms, paving the way for further simplification using other mathematical principles. With the quotient rule in our arsenal, we are well-equipped to tackle the challenges posed by the expression.

Simplifying the Radicand

Inside the radical, we have the expression ${\frac{14 x^8}{7 x^4}}$. Let's simplify this fraction by dividing the coefficients and applying the quotient rule for exponents. The quotient rule for exponents states that when dividing exponential terms with the same base, we subtract the exponents. Mathematically, this can be expressed as $\frac{xm}{xn} = x^{m-n}$, where $x \neq 0$ and $m$ and $n$ are real numbers. Applying this rule to our expression, we can simplify the variable terms within the radical. By dividing the coefficients and subtracting the exponents, we can reduce the radicand to a simpler form. This simplification is a crucial step in our quest to find the equivalent expression. A simpler radicand makes it easier to extract any perfect square factors, further simplifying the radical expression. With the quotient rule for exponents at our disposal, we can effectively manipulate the radicand and bring us closer to the final solution.

Extracting Perfect Squares

After simplifying the radicand, we obtain an expression that involves a perfect square. Extracting perfect squares from a radical is a key step in simplifying radical expressions. A perfect square is a number or expression that can be obtained by squaring another number or expression. For example, 4 is a perfect square because it is the square of 2, and $x^2$ is a perfect square because it is the square of $x$. When we encounter a perfect square within a radical, we can take its square root and move it outside the radical. This process simplifies the radical expression and brings us closer to the final answer. In our case, we can identify a perfect square within the simplified radicand and extract it, further simplifying the expression. This step is crucial in revealing the equivalent form of the original expression.

Putting It All Together

Now that we have simplified the radical and the expression, let's put it all together. We started with the expression $18 x^2 \sqrt{14 x^8} \div 6 \sqrt{7 x^4}$ and systematically applied mathematical principles to simplify it. We used the quotient rule for radicals, the quotient rule for exponents, and the concept of extracting perfect squares. By carefully applying these principles, we have transformed the expression into a simpler form. Now, it's time to combine the simplified components and arrive at the final answer. This step involves multiplying the coefficients and combining the variable terms. By meticulously performing these operations, we can unveil the equivalent form of the original expression and complete our mathematical journey. The final result will be a simplified expression that is easier to understand and work with.

The Final Equivalent Expression

After meticulously applying mathematical principles, we arrive at the final equivalent expression: $3 x^4 \sqrt{2}$. This expression is the simplified form of the original expression, $18 x^2 \sqrt{14 x^8} \div 6 \sqrt{7 x^4}$. The journey to simplification involved breaking down the expression into its components, applying the quotient rule for radicals, simplifying the radicand, extracting perfect squares, and combining the simplified terms. Each step was crucial in transforming the expression into its equivalent form. The final expression, $3 x^4 \sqrt{2}$, is not only simpler but also reveals the underlying structure of the original expression. This process of simplification highlights the power of mathematical principles in unraveling complex expressions and revealing their inherent simplicity. The final answer demonstrates our ability to navigate through algebraic intricacies and arrive at a concise and elegant solution.

Conclusion

In conclusion, the expression $18 x^2 \sqrt{14 x^8} \div 6 \sqrt{7 x^4}$, where $x \neq 0$, is equivalent to $3 x^4 \sqrt{2}$. This simplification was achieved by systematically applying mathematical principles, including the quotient rule for radicals, the quotient rule for exponents, and the concept of extracting perfect squares. The process of simplifying algebraic expressions is not merely about finding the right answer; it's about developing a deep understanding of mathematical principles and their applications. By breaking down complex expressions into smaller, manageable components, we can apply these principles to transform them into simpler forms. This journey of simplification enhances our problem-solving skills and deepens our appreciation for the elegance and power of mathematics. The final equivalent expression, $3 x^4 \sqrt{2}$, stands as a testament to our ability to navigate through algebraic intricacies and arrive at a concise and meaningful solution. This exercise not only reinforces our understanding of mathematical concepts but also empowers us to tackle future challenges with confidence and precision.

Therefore, the correct answer is B. $3 x^4 \sqrt{2}$.