Simplifying Radical Expressions A Step-by-Step Guide To \$\frac{\sqrt{72 X^3}}{\sqrt{162 X}}$

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Simplify the expression $\frac{\sqrt{72 x^3}}{\sqrt{162 x}}$. Assume all variables are positive.

In the realm of mathematics, simplifying expressions is a fundamental skill. It allows us to represent mathematical concepts in their most concise and understandable form. Today, we will delve into simplifying a radical expression, specifically, $\frac{\sqrt{72 x^3}}{\sqrt{162 x}}. This exercise is not merely about manipulating symbols; it's about understanding the underlying principles of radicals, exponents, and factorization. By the end of this article, you'll have a comprehensive grasp of how to tackle such problems with confidence and precision.

Breaking Down the Problem

To effectively simplify $\frac{\sqrt{72 x^3}}{\sqrt{162 x}}$, we'll employ several key strategies. First, we'll utilize the property of radicals that states $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$, where a and b are non-negative and b is not zero. This allows us to combine the two radicals into one, making the expression more manageable. Second, we'll factor the numbers under the radical sign into their prime factors. This will reveal any perfect square factors that can be extracted from the radical. Third, we'll apply the rules of exponents to simplify the variable terms. Remember that $\sqrt{x^2} = |x|$ , but since we're assuming all variables are positive, we can simply write $\sqrt{x^2} = x$. By systematically applying these techniques, we'll arrive at the simplified form of the expression.

Step-by-Step Simplification

Let's embark on the step-by-step simplification process:

  1. Combine the Radicals:

    Using the property $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$, we rewrite the expression as:

    72x3162x\sqrt{\frac{72 x^3}{162 x}}

    This step consolidates the two radicals into one, setting the stage for further simplification.

  2. Simplify the Fraction Inside the Radical:

    Before dealing with the square root, let's simplify the fraction $\frac72 x^3}{162 x}$. We can simplify the numerical part by finding the greatest common divisor (GCD) of 72 and 162, which is 18. Dividing both numerator and denominator by 18, we get $\frac{72}{162} = \frac{4}{9}$. For the variable part, we use the rule of exponents $\frac{x^3{x} = x^{3-1} = x^2$. Thus, the fraction simplifies to:

    72x3162x=4x29\frac{72 x^3}{162 x} = \frac{4x^2}{9}

    Now, our expression looks like:

    4x29\sqrt{\frac{4x^2}{9}}

    This simplified fraction under the radical makes the next step easier.

  3. Take the Square Root:

    Now we can take the square root of the simplified fraction. We know that $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$. Applying this, we have:

    4x29=4x29\sqrt{\frac{4x^2}{9}} = \frac{\sqrt{4x^2}}{\sqrt{9}}

    We know that $\sqrt{4} = 2$, $\sqrt{x^2} = x$, and $\sqrt{9} = 3$. Therefore,

    4x29=2x3\frac{\sqrt{4x^2}}{\sqrt{9}} = \frac{2x}{3}

    And that's it! We've successfully simplified the radical expression.

Final Answer

The simplified form of $\frac{\sqrt{72 x^3}}{\sqrt{162 x}}$ is $\frac{2x}{3}$.

Common Mistakes to Avoid

When simplifying radical expressions, there are several common pitfalls to watch out for. One frequent mistake is failing to completely factor the numbers under the radical. Always ensure you've extracted the largest possible perfect square factors. Another error is incorrectly applying the rules of exponents. Remember that $\sqrt{x^2} = |x|$ , and when dealing with variables, pay close attention to the exponents. Lastly, be mindful of the order of operations. Simplify the fraction inside the radical before attempting to take the square root. By being aware of these potential pitfalls, you can improve your accuracy and avoid unnecessary errors.

Practice Problems

To solidify your understanding, let's tackle a few practice problems:

  1. Simplify $\frac{\sqrt{50 a^5}}{\sqrt{8 a}}$
  2. Simplify $\frac{\sqrt{27 b^7}}{\sqrt{12 b^3}}$
  3. Simplify $\sqrt{\frac{98 x^9}{2 x}}$

Working through these problems will reinforce the steps we've discussed and build your confidence in simplifying radical expressions. Remember to break down the problem into manageable steps, focus on prime factorization, and apply the rules of exponents correctly. The more you practice, the more proficient you'll become.

Conclusion

Simplifying radical expressions is a crucial skill in mathematics, and mastering it requires a solid understanding of radicals, exponents, and factorization. By following a systematic approach, such as the one outlined in this article, you can confidently tackle even complex expressions. Remember to break down the problem, factor completely, apply the rules of exponents, and double-check your work. With practice, you'll find that simplifying radicals becomes second nature, empowering you to excel in your mathematical endeavors. The simplified form of the given expression $\frac{\sqrt{72 x^3}}{\sqrt{162 x}}$ is $\frac{2x}{3}$, demonstrating the power of mathematical simplification.