Which Expression Is Equivalent To The Product Of Radicals When X > 0? $\sqrt{\frac{2x}{3}} \cdot \sqrt{\frac{x}{6}}$

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sqrtfrac2x3cdotsqrtfracx6\\sqrt{\\frac{2x}{3}} \\cdot \\sqrt{\\frac{x}{6}}

A. fracsqrtx3\\frac{\\sqrt{x}}{3}
B. fracx9\\frac{x}{9}
C. fracsqrt2x9\\frac{\\sqrt{2x}}{9}

This detailed guide aims to provide a thorough understanding of simplifying radical expressions, specifically addressing the problem of finding an equivalent expression for the product 2x3⋅x6\sqrt{\frac{2x}{3}} \cdot \sqrt{\frac{x}{6}} when x>0x > 0. This type of problem is a staple in algebra, often appearing in standardized tests and serving as a fundamental concept for more advanced mathematical topics. We will delve into the step-by-step process of simplifying the given expression, highlighting the properties of radicals and exponents that are crucial for solving it. Furthermore, we will explore common pitfalls and strategies to ensure accuracy and efficiency in solving similar problems. Understanding these concepts is not only beneficial for academic success but also for developing a strong foundation in mathematical reasoning.

Breaking Down the Problem

To effectively tackle this problem, we need to understand the properties of radicals and how they interact with multiplication and division. Radicals, often represented by the square root symbol (\sqrt{}), indicate a root of a number. In this case, we are dealing with square roots, which means we are looking for a number that, when multiplied by itself, equals the number under the radical. The expression a\sqrt{a} represents the principal square root of aa. When multiplying radicals, a key property comes into play: a⋅b=a⋅b\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}. This property allows us to combine the two radicals in our problem into a single radical, which is the first step towards simplification. By applying this property, we transform the initial expression into a more manageable form, setting the stage for further simplification and ultimately leading us to the equivalent expression. The ability to manipulate radicals in this way is essential for solving a wide range of algebraic problems.

Step-by-Step Solution

Let's walk through the solution step-by-step to ensure clarity and understanding.

  1. Combine the Radicals: Using the property a⋅b=a⋅b\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}, we can combine the two radicals:

    2x3cdotsqrtx6=sqrt2x3cdotfracx6\sqrt{\frac{2x}{3}} \\cdot \\sqrt{\frac{x}{6}} = \\sqrt{\frac{2x}{3} \\cdot \\frac{x}{6}}

    This step simplifies the expression by bringing the two separate radicals under one square root, making the subsequent steps easier to perform.

  2. Multiply the Fractions: Now, we multiply the fractions inside the square root:

    sqrt2x3cdotfracx6=sqrt2x218\\sqrt{\frac{2x}{3} \\cdot \\frac{x}{6}} = \\sqrt{\frac{2x^2}{18}}

    Here, we multiply the numerators (2x and x) and the denominators (3 and 6) separately. This results in a single fraction under the radical, which is a crucial step in simplifying the expression.

  3. Simplify the Fraction: Simplify the fraction inside the square root by reducing it to its simplest form:

    sqrt2x218=sqrtx29\\sqrt{\frac{2x^2}{18}} = \\sqrt{\frac{x^2}{9}}

    We divide both the numerator and the denominator by their greatest common divisor, which is 2. This simplification makes it easier to extract the square root in the next step.

  4. Take the Square Root: Now, we take the square root of both the numerator and the denominator:

    sqrtx29=fracsqrtx2sqrt9=fracx3\\sqrt{\frac{x^2}{9}} = \\frac{\\sqrt{x^2}}{\\sqrt{9}} = \\frac{x}{3}

    Since x>0x > 0, the square root of x2x^2 is simply xx. The square root of 9 is 3. This step effectively removes the radical, leaving us with a simplified algebraic expression.

  5. Find the Equivalent Choice: However, x3\frac{x}{3} is not one of the options provided. It seems there was a miscalculation in the provided solution. Let's re-evaluate from step 3:

    sqrtx29=fracsqrtx2sqrt9=frac∣x∣3\\sqrt{\frac{x^2}{9}} = \\frac{\\sqrt{x^2}}{\\sqrt{9}} = \\frac{|x|}{3}

    Since we know x>0x > 0, then ∣x∣=x|x| = x, so this part is correct. But let's backtrack and see if we made a mistake before that. Going back to step 3:

    sqrt2x218=sqrtx29\\sqrt{\frac{2x^2}{18}} = \\sqrt{\frac{x^2}{9}}

    This step is correct. Let's re-evaluate step 4. The mistake we made is that after simplifying the fraction, we incorrectly took the square root. We should have:

sqrtfracx29=fracsqrtx2sqrt9=fracx3\\sqrt{\\frac{x^2}{9}} = \\frac{\\sqrt{x^2}}{\\sqrt{9}} = \\frac{x}{3}

It appears there's an issue with the provided answer choices because x3\frac{x}{3} isn't an option. Let's re-examine our work and the original problem to ensure we haven't missed anything. We combined the radicals, multiplied the fractions, simplified the fraction, and took the square root. All steps seem logically sound.

Given the options:

A. fracsqrtx3\\frac{\\sqrt{x}}{3} B. fracx3\\frac{x}{3} C. fracsqrt2x9\\frac{\\sqrt{2x}}{9}

Our calculated answer, x3\frac{x}{3}, still doesn't match any of these. It's possible there's a typo in the options or the original problem statement. However, based on our step-by-step simplification, x3\frac{x}{3} is the correct simplified form of the given expression. So option B x9\frac{x}{9} is likely the one that was intended to be the answer. There could have been a typo in the question, so the correct answer is closest to A. x3\frac{\sqrt{x}}{3} if we assume that was intended to be x3\frac{x}{3}.

Common Mistakes and How to Avoid Them

When working with radical expressions, there are several common mistakes that students often make. Understanding these pitfalls can help you avoid them and improve your accuracy.

  1. Incorrectly Applying the Distributive Property: One common mistake is attempting to distribute a radical over addition or subtraction. For example, a+b\sqrt{a + b} is not equal to a+b\sqrt{a} + \sqrt{b}. This is a critical error that can lead to incorrect simplifications. To avoid this, always remember that the distributive property does not apply to radicals in this way. Instead, focus on simplifying the expression under the radical first, if possible, before attempting to take the square root.

  2. Forgetting to Simplify Radicals Completely: Another frequent error is not simplifying radicals to their simplest form. A radical is considered fully simplified when the radicand (the number under the radical) has no perfect square factors other than 1. For instance, 8\sqrt{8} should be simplified to 222\sqrt{2}. To ensure complete simplification, factor the radicand and look for pairs of identical factors. Each pair can be brought outside the radical as a single factor. This process ensures that the radical is in its most reduced form.

  3. Misunderstanding the Properties of Radicals: A lack of understanding of the fundamental properties of radicals can lead to errors. For example, the property a⋅b=a⋅b\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} is crucial for multiplying radicals, but it must be applied correctly. Similarly, when dividing radicals, the property ab=ab\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} is essential. Make sure you are comfortable with these properties and how to apply them in different situations. Practice using these properties in various problems to solidify your understanding.

  4. Ignoring the Domain of the Variable: When dealing with variables under radicals, it's crucial to consider the domain of the variable. For example, x2\sqrt{x^2} is equal to ∣x∣|x|, not simply xx. If xx is negative, then x2\sqrt{x^2} would be −x-x. In the given problem, we were told that x>0x > 0, which simplifies the situation, but it's important to be mindful of this in general. Always consider the possible values of the variable and how they affect the expression.

Practice Problems

To further solidify your understanding, here are some practice problems similar to the one we solved. Work through these problems step-by-step, paying attention to the properties of radicals and the common mistakes discussed earlier.

  1. Simplify: 3x4⋅x12\sqrt{\frac{3x}{4}} \cdot \sqrt{\frac{x}{12}} when x>0x > 0
  2. Simplify: 5x28⋅2x\sqrt{\frac{5x^2}{8}} \cdot \sqrt{\frac{2}{x}} when x>0x > 0
  3. Simplify: 4x9⋅x16\sqrt{\frac{4x}{9}} \cdot \sqrt{\frac{x}{16}} when x>0x > 0

Working through these problems will help you become more comfortable with simplifying radical expressions and build confidence in your problem-solving abilities. Remember to break down each problem into smaller steps, apply the properties of radicals correctly, and double-check your work to avoid common mistakes.

Conclusion

In conclusion, simplifying radical expressions involves a systematic application of the properties of radicals and careful attention to detail. By understanding the fundamental principles and practicing regularly, you can master this skill and excel in algebra and beyond. Remember to combine radicals using the appropriate properties, simplify fractions under the radical, and take the square root of both the numerator and the denominator when possible. Be mindful of common mistakes, such as incorrectly applying the distributive property or forgetting to simplify radicals completely. By following these guidelines and practicing consistently, you will be well-equipped to tackle any radical simplification problem that comes your way. The key to success lies in a solid understanding of the underlying concepts and a commitment to careful, step-by-step problem-solving.