Find The Simplified Product: 2 X 3 ⋅ 18 X 5 \sqrt{2 X^3} \cdot \sqrt{18 X^5} 2 X 3 ​ ⋅ 18 X 5 ​ A. 6 X 4 \sqrt{6 X^4} 6 X 4 ​ B. 36 X 8 \sqrt{36 X^8} 36 X 8 ​ C. 18 X 4 18 X^4 18 X 4 D. 6 X 4 6 X^4 6 X 4

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Understanding the Problem

When dealing with square roots, it's essential to remember that the square root of a product is equal to the product of the square roots. This property can be expressed as: $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$. In this problem, we are given the expression $\sqrt{2 x^3} \cdot \sqrt{18 x^5}$, and we need to simplify it.

Breaking Down the Expression

To simplify the given expression, we need to first break it down into its prime factors. The expression $\sqrt{2 x^3} \cdot \sqrt{18 x^5}$ can be rewritten as $\sqrt{2} \cdot \sqrt{x^3} \cdot \sqrt{18} \cdot \sqrt{x^5}$.

Simplifying the Square Roots

Now, let's simplify each square root individually. The square root of a number can be simplified by finding the largest perfect square that divides the number. For example, $\sqrt{18}$ can be simplified as $\sqrt{9 \cdot 2}$, which is equal to $3\sqrt{2}$.

Applying the Property of Square Roots

Using the property of square roots mentioned earlier, we can rewrite the expression as $\sqrt{2} \cdot \sqrt{x^3} \cdot 3\sqrt{2} \cdot \sqrt{x^5}$. Now, we can combine the like terms, which are the square roots of $2$ and the square roots of $x$.

Simplifying the Expression

Combining the like terms, we get $\sqrt{2} \cdot 3\sqrt{2} \cdot \sqrt{x^3} \cdot \sqrt{x^5}$. This can be further simplified as $3 \cdot 2 \cdot \sqrt{x^3} \cdot \sqrt{x^5}$, which is equal to $6 \cdot \sqrt{x^3} \cdot \sqrt{x^5}$.

Simplifying the Square Roots of x

Now, let's simplify the square roots of $x$. The square root of $x^3$ can be simplified as $x^{3/2}$, and the square root of $x^5$ can be simplified as $x^{5/2}$.

Combining the Terms

Combining the terms, we get $6 \cdot x^{3/2} \cdot x^{5/2}$. Using the property of exponents, we can add the exponents, which gives us $6 \cdot x^{8/2}$.

Simplifying the Expression

Simplifying the expression further, we get $6 \cdot x^4$. Therefore, the simplified product of $\sqrt{2 x^3} \cdot \sqrt{18 x^5}$ is $6 x^4$.

Conclusion

In conclusion, the simplified product of $\sqrt{2 x^3} \cdot \sqrt{18 x^5}$ is $6 x^4$. This can be verified by multiplying the two expressions together and simplifying the result.

Final Answer

The final answer is $\boxed{6 x^4}$.

Discussion

This problem requires the application of the property of square roots, which states that the square root of a product is equal to the product of the square roots. It also requires the simplification of square roots and the application of the property of exponents.

Related Problems

This problem is related to other problems that involve the simplification of square roots and the application of the property of exponents. Some examples of related problems include:

• Simplifying the expression $\sqrt{3 x^2} \cdot \sqrt{4 x^3}$
• Simplifying the expression $\sqrt{5 x^4} \cdot \sqrt{9 x^2}$
• Simplifying the expression $\sqrt{7 x^5} \cdot \sqrt{2 x^3}$

Practice Problems

To practice solving problems like this, try simplifying the following expressions:

• $\sqrt{2 x^3} \cdot \sqrt{12 x^5}$
• $\sqrt{3 x^2} \cdot \sqrt{8 x^4}$
• $\sqrt{4 x^5} \cdot \sqrt{9 x^3}$

Tips and Tricks

When simplifying expressions involving square roots, remember to:

• Break down the expression into its prime factors
• Simplify each square root individually
• Apply the property of square roots to combine the like terms
• Simplify the expression further by combining the terms

By following these tips and tricks, you can simplify expressions involving square roots and apply the property of exponents to solve problems like this.

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Q: What is the property of square roots that we need to apply in this problem?

A: The property of square roots that we need to apply in this problem is that the square root of a product is equal to the product of the square roots. This can be expressed as: $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$.

Q: How do we simplify the square roots in this problem?

A: To simplify the square roots in this problem, we need to first break them down into their prime factors. For example, $\sqrt{18}$ can be simplified as $\sqrt{9 \cdot 2}$, which is equal to $3\sqrt{2}$.

Q: What is the simplified form of $\sqrt{2 x^3} \cdot \sqrt{18 x^5}$?

A: The simplified form of $\sqrt{2 x^3} \cdot \sqrt{18 x^5}$ is $6 x^4$.

Q: How do we combine the like terms in this problem?

A: To combine the like terms in this problem, we need to apply the property of square roots, which states that the square root of a product is equal to the product of the square roots. This allows us to combine the like terms, such as the square roots of $2$ and the square roots of $x$.

Q: What is the final answer to this problem?

A: The final answer to this problem is $\boxed{6 x^4}$.

Q: What are some related problems that involve the simplification of square roots and the application of the property of exponents?

A: Some related problems that involve the simplification of square roots and the application of the property of exponents include:

• Simplifying the expression $\sqrt{3 x^2} \cdot \sqrt{4 x^3}$
• Simplifying the expression $\sqrt{5 x^4} \cdot \sqrt{9 x^2}$
• Simplifying the expression $\sqrt{7 x^5} \cdot \sqrt{2 x^3}$

Q: How can we practice solving problems like this?

A: To practice solving problems like this, try simplifying the following expressions:

• $\sqrt{2 x^3} \cdot \sqrt{12 x^5}$
• $\sqrt{3 x^2} \cdot \sqrt{8 x^4}$
• $\sqrt{4 x^5} \cdot \sqrt{9 x^3}$

Q: What are some tips and tricks for simplifying expressions involving square roots?

A: Some tips and tricks for simplifying expressions involving square roots include:

• Breaking down the expression into its prime factors
• Simplifying each square root individually
• Applying the property of square roots to combine the like terms
• Simplifying the expression further by combining the terms

Q: Why is it important to understand the property of square roots?

A: Understanding the property of square roots is important because it allows us to simplify expressions involving square roots and apply the property of exponents to solve problems like this.

Q: Can you provide more examples of problems that involve the simplification of square roots and the application of the property of exponents?

A: Yes, here are some more examples of problems that involve the simplification of square roots and the application of the property of exponents:

• Simplifying the expression $\sqrt{6 x^3} \cdot \sqrt{9 x^5}$
• Simplifying the expression $\sqrt{8 x^2} \cdot \sqrt{12 x^4}$
• Simplifying the expression $\sqrt{10 x^5} \cdot \sqrt{4 x^3}$

Q: How can we use the property of square roots to solve problems in real-life situations?

A: The property of square roots can be used to solve problems in real-life situations, such as calculating the area of a square or the volume of a cube. For example, if we need to calculate the area of a square with side length $x$, we can use the formula $A = x^2$, which involves the square root of $x$.