Simplifying Radical Expressions A Step By Step Guide

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In this comprehensive guide, we will delve into the process of simplifying radical expressions and identifying equivalent forms. We will specifically address the expression $2x\sqrt{44x} - 2\sqrt{11x^3}$, where $x > 0$, and determine which of the provided options (A. $2x\sqrt{11x}$, B. $6x\sqrt{22x}$, C. $2x$, D. $8x^2\sqrt{11}$) is equivalent. This exploration will enhance your understanding of radical simplification and algebraic manipulation. Let's embark on this mathematical journey together!

Simplifying the Expression

To find the equivalent expression, we need to simplify the given expression: $2x\sqrt{44x} - 2\sqrt{11x^3}$. The key to simplifying such expressions lies in understanding how to manipulate radicals and combine like terms. Let's break down the simplification process step by step.

First, let's focus on the term $2x\sqrt{44x}$. We can simplify the radical part, $\sqrt{44x}$, by factoring out perfect squares from 44. Since $44 = 4 \times 11$, we can rewrite the radical as follows:

$\sqrt{44x} = \sqrt{4 \times 11 \times x} = \sqrt{4} \times \sqrt{11x} = 2\sqrt{11x}$

Now, substituting this back into the term, we get:

$2x\sqrt{44x} = 2x(2\sqrt{11x}) = 4x\sqrt{11x}$

Next, let's simplify the second term, $2\sqrt{11x^3}$. Here, we need to consider the $x^3$ inside the square root. We can rewrite $x^3$ as $x^2 \times x$, where $x^2$ is a perfect square. Thus, we have:

$2\sqrt{11x^3} = 2\sqrt{11 \times x^2 \times x} = 2\sqrt{x^2} \times \sqrt{11x} = 2x\sqrt{11x}$

Now that we have simplified both terms, we can rewrite the original expression:

$2x\sqrt{44x} - 2\sqrt{11x^3} = 4x\sqrt{11x} - 2x\sqrt{11x}$

Notice that both terms now have the same radical part, $\sqrt{11x}$. This means we can combine these terms by subtracting their coefficients:

$4x\sqrt{11x} - 2x\sqrt{11x} = (4x - 2x)\sqrt{11x} = 2x\sqrt{11x}$

Therefore, the simplified expression is $2x\sqrt{11x}$.

Evaluating the Options

Now that we have simplified the expression, let's compare our result with the provided options:

A. $2x\sqrt{11x}$ B. $6x\sqrt{22x}$ C. $2x$ D. $8x^2\sqrt{11}$

By comparing our simplified expression, $2x\sqrt{11x}$, with the options, we can see that option A, $2x\sqrt{11x}$, matches our result. Therefore, option A is the equivalent expression.

Options B, C, and D do not match our simplified expression. Option B, $6x\sqrt{22x}$, has a different constant and radical part. Option C, $2x$, does not have the radical part. Option D, $8x^2\sqrt{11}$, has a different power of $x$ and a different constant inside the radical.

Thus, the correct answer is option A.

Detailed Explanation of Each Step

To ensure a thorough understanding, let's revisit each step of the simplification process with a more detailed explanation.

Step 1: Simplify $2x\sqrt{44x}$

We begin by focusing on the first term, $2x\sqrt{44x}$. The goal here is to simplify the radical part, $\sqrt{44x}$.

Factoring 44

The first step in simplifying $\sqrt{44x}$ is to factor 44 into its prime factors. We find that $44 = 2 \times 2 \times 11 = 2^2 \times 11$. This factorization helps us identify any perfect square factors.

Rewriting the Radical

Now we can rewrite $\sqrt{44x}$ as $\sqrt{2^2 \times 11 \times x}$. This allows us to separate the perfect square factor ($2^2$) from the rest of the terms.

Applying the Product Rule for Radicals

The product rule for radicals states that $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$. Applying this rule, we get:

$\sqrt{2^2 \times 11 \times x} = \sqrt{2^2} \times \sqrt{11x}$

Simplifying the Perfect Square

Since $\sqrt{2^2} = 2$, we can simplify further:

$\sqrt{2^2} \times \sqrt{11x} = 2\sqrt{11x}$

Substituting Back into the Term

Now we substitute this back into the original term:

$2x\sqrt{44x} = 2x(2\sqrt{11x}) = 4x\sqrt{11x}$

Step 2: Simplify $2\sqrt{11x^3}$

Next, we focus on the second term, $2\sqrt{11x^3}$. Here, we need to simplify the radical part, $\sqrt{11x^3}$.

Factoring $x^3$

We can rewrite $x^3$ as $x^2 \times x$. This is because $x^2$ is a perfect square, which we can simplify within the radical.

Rewriting the Radical

Now we rewrite $\sqrt{11x^3}$ as $\sqrt{11 \times x^2 \times x}$. This allows us to separate the perfect square factor ($x^2$) from the rest of the terms.

Applying the Product Rule for Radicals

Applying the product rule for radicals, we get:

$\sqrt{11 \times x^2 \times x} = \sqrt{x^2} \times \sqrt{11x}$

Simplifying the Perfect Square

Since $\sqrt{x^2} = x$ (because $x > 0$), we can simplify further:

$\sqrt{x^2} \times \sqrt{11x} = x\sqrt{11x}$

Substituting Back into the Term

Now we substitute this back into the original term:

$2\sqrt{11x^3} = 2(x\sqrt{11x}) = 2x\sqrt{11x}$

Step 3: Combine the Simplified Terms

Now that we have simplified both terms, we can rewrite the original expression:

$2x\sqrt{44x} - 2\sqrt{11x^3} = 4x\sqrt{11x} - 2x\sqrt{11x}$

Combining Like Terms

Notice that both terms now have the same radical part, $\sqrt{11x}$. This means we can combine these terms by subtracting their coefficients:

$4x\sqrt{11x} - 2x\sqrt{11x} = (4x - 2x)\sqrt{11x}$

Final Simplification

Subtracting the coefficients, we get:

$(4x - 2x)\sqrt{11x} = 2x\sqrt{11x}$

Thus, the simplified expression is $2x\sqrt{11x}$.

Conclusion

In conclusion, the expression equivalent to $2x\sqrt{44x} - 2\sqrt{11x^3}$, given that $x > 0$, is $2x\sqrt{11x}$. This was determined by simplifying each term in the expression separately and then combining like terms. The key steps involved factoring the numbers and variables inside the radicals, applying the product rule for radicals, and simplifying perfect squares. This detailed explanation should provide a clear understanding of the process and the reasoning behind each step.

Understanding how to simplify radical expressions is a fundamental skill in algebra. By mastering these techniques, you can confidently tackle more complex problems involving radicals. Remember to always look for perfect square factors within the radicals and use the properties of radicals to simplify expressions effectively. This will not only help you in your mathematical studies but also in various practical applications where radical simplification is necessary.

By carefully breaking down each step and providing detailed explanations, we have shown how to simplify the given expression and identify the correct equivalent form. This approach not only helps in solving this particular problem but also builds a strong foundation for handling similar mathematical challenges in the future. The ability to simplify expressions and manipulate radicals is crucial for success in algebra and beyond.