What Is The Following Product? Assume $x \geq 0$ And $y \geq 0$.$\sqrt{5 X^8 Y^2} \cdot \sqrt{10 X^3} \cdot \sqrt{12 Y}$A. $3 X^5 Y \sqrt{3 X Y}$ B. $10 X^5 Y \sqrt{6 X Y}$ C. $3 X^3 Y \sqrt{3 X^2

Understanding the Problem

The given problem involves simplifying a mathematical expression that includes square roots and variables. We are asked to assume that both $x$ and $y$ are non-negative numbers, denoted as $x \geq 0$ and $y \geq 0$. Our goal is to simplify the expression $\sqrt{5 x^8 y^2} \cdot \sqrt{10 x^3} \cdot \sqrt{12 y}$ and determine the correct answer among the given options.

Breaking Down the Expression

To simplify the given expression, we can start by breaking it down into smaller parts. We can rewrite the expression as follows:

$\sqrt{5 x^8 y^2} \cdot \sqrt{10 x^3} \cdot \sqrt{12 y}$

$= \sqrt{5} \cdot \sqrt{x^8} \cdot \sqrt{y^2} \cdot \sqrt{10} \cdot \sqrt{x^3} \cdot \sqrt{12} \cdot \sqrt{y}$

$= \sqrt{5} \cdot x^4 \cdot y \cdot \sqrt{10} \cdot x^{3/2} \cdot \sqrt{12} \cdot y^{1/2}$

Simplifying the Square Roots

Now that we have broken down the expression, we can simplify the square roots. We know that $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$, so we can rewrite the expression as follows:

$= \sqrt{5} \cdot x^4 \cdot y \cdot \sqrt{10} \cdot x^{3/2} \cdot \sqrt{12} \cdot y^{1/2}$

$= \sqrt{5} \cdot x^4 \cdot y \cdot \sqrt{10} \cdot x^{3/2} \cdot 2 \sqrt{3} \cdot y^{1/2}$

$= \sqrt{5} \cdot x^4 \cdot y \cdot \sqrt{10} \cdot x^{3/2} \cdot 2 \sqrt{3} \cdot y^{1/2}$

Combining Like Terms

Now that we have simplified the square roots, we can combine like terms. We can rewrite the expression as follows:

$= \sqrt{5} \cdot x^4 \cdot y \cdot \sqrt{10} \cdot x^{3/2} \cdot 2 \sqrt{3} \cdot y^{1/2}$

$= \sqrt{5} \cdot x^{4+3/2} \cdot y^{1+1/2} \cdot \sqrt{10} \cdot 2 \sqrt{3}$

$= \sqrt{5} \cdot x^{7/2} \cdot y^{3/2} \cdot \sqrt{10} \cdot 2 \sqrt{3}$

Simplifying the Expression

Now that we have combined like terms, we can simplify the expression further. We can rewrite the expression as follows:

$= \sqrt{5} \cdot x^{7/2} \cdot y^{3/2} \cdot \sqrt{10} \cdot 2 \sqrt{3}$

$= \sqrt{5} \cdot x^{7/2} \cdot y^{3/2} \cdot \sqrt{10 \cdot 12}$

$= \sqrt{5} \cdot x^{7/2} \cdot y^{3/2} \cdot \sqrt{120}$

$= \sqrt{5} \cdot x^{7/2} \cdot y^{3/2} \cdot \sqrt{4 \cdot 30}$

$= \sqrt{5} \cdot x^{7/2} \cdot y^{3/2} \cdot 2 \sqrt{30}$

$= 2 \sqrt{5} \cdot x^{7/2} \cdot y^{3/2} \cdot \sqrt{30}$

$= 2 \sqrt{5} \cdot x^{7/2} \cdot y^{3/2} \cdot \sqrt{2 \cdot 15}$

$= 2 \sqrt{5} \cdot x^{7/2} \cdot y^{3/2} \cdot \sqrt{2} \cdot \sqrt{15}$

$= 2 \sqrt{5} \cdot x^{7/2} \cdot y^{3/2} \cdot \sqrt{2} \cdot \sqrt{3} \cdot \sqrt{5}$

$= 2 \cdot 5 \cdot x^{7/2} \cdot y^{3/2} \cdot \sqrt{2} \cdot \sqrt{3}$

$= 10 \cdot x^{7/2} \cdot y^{3/2} \cdot \sqrt{6}$

$= 10 \cdot x^{7/2} \cdot y^{3/2} \cdot \sqrt{6}$

Evaluating the Options

Now that we have simplified the expression, we can evaluate the options. We can rewrite the expression as follows:

$= 10 \cdot x^{7/2} \cdot y^{3/2} \cdot \sqrt{6}$

This expression matches option B, which is $10 x^5 y \sqrt{6 x y}$. Therefore, the correct answer is option B.

Conclusion

In conclusion, we have simplified the given expression and determined the correct answer among the given options. We started by breaking down the expression into smaller parts and simplifying the square roots. We then combined like terms and simplified the expression further. Finally, we evaluated the options and determined that the correct answer is option B, which is $10 x^5 y \sqrt{6 x y}$.

Final Answer

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Q: What is the final answer to the expression $\sqrt{5 x^8 y^2} \cdot \sqrt{10 x^3} \cdot \sqrt{12 y}$?

A: The final answer is $10 x^5 y \sqrt{6 x y}$.

Q: How do you simplify the expression $\sqrt{5 x^8 y^2} \cdot \sqrt{10 x^3} \cdot \sqrt{12 y}$?

A: To simplify the expression, you can start by breaking it down into smaller parts. You can rewrite the expression as follows:

$\sqrt{5 x^8 y^2} \cdot \sqrt{10 x^3} \cdot \sqrt{12 y}$

$= \sqrt{5} \cdot \sqrt{x^8} \cdot \sqrt{y^2} \cdot \sqrt{10} \cdot \sqrt{x^3} \cdot \sqrt{12} \cdot \sqrt{y}$

$= \sqrt{5} \cdot x^4 \cdot y \cdot \sqrt{10} \cdot x^{3/2} \cdot \sqrt{12} \cdot y^{1/2}$

You can then simplify the square roots and combine like terms to get the final answer.

Q: What is the first step in simplifying the expression $\sqrt{5 x^8 y^2} \cdot \sqrt{10 x^3} \cdot \sqrt{12 y}$?

A: The first step is to break down the expression into smaller parts. You can rewrite the expression as follows:

$\sqrt{5 x^8 y^2} \cdot \sqrt{10 x^3} \cdot \sqrt{12 y}$

$= \sqrt{5} \cdot \sqrt{x^8} \cdot \sqrt{y^2} \cdot \sqrt{10} \cdot \sqrt{x^3} \cdot \sqrt{12} \cdot \sqrt{y}$

Q: How do you simplify the square roots in the expression $\sqrt{5 x^8 y^2} \cdot \sqrt{10 x^3} \cdot \sqrt{12 y}$?

A: To simplify the square roots, you can use the property that $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$. You can rewrite the expression as follows:

$= \sqrt{5} \cdot x^4 \cdot y \cdot \sqrt{10} \cdot x^{3/2} \cdot \sqrt{12} \cdot y^{1/2}$

$= \sqrt{5} \cdot x^4 \cdot y \cdot \sqrt{10} \cdot x^{3/2} \cdot 2 \sqrt{3} \cdot y^{1/2}$

Q: How do you combine like terms in the expression $\sqrt{5 x^8 y^2} \cdot \sqrt{10 x^3} \cdot \sqrt{12 y}$?

A: To combine like terms, you can rewrite the expression as follows:

$= \sqrt{5} \cdot x^4 \cdot y \cdot \sqrt{10} \cdot x^{3/2} \cdot 2 \sqrt{3} \cdot y^{1/2}$

$= \sqrt{5} \cdot x^{4+3/2} \cdot y^{1+1/2} \cdot \sqrt{10} \cdot 2 \sqrt{3}$

$= \sqrt{5} \cdot x^{7/2} \cdot y^{3/2} \cdot \sqrt{10} \cdot 2 \sqrt{3}$

Q: What is the final simplified expression for $\sqrt{5 x^8 y^2} \cdot \sqrt{10 x^3} \cdot \sqrt{12 y}$?

A: The final simplified expression is $10 x^5 y \sqrt{6 x y}$.

Q: How do you evaluate the options for the expression $\sqrt{5 x^8 y^2} \cdot \sqrt{10 x^3} \cdot \sqrt{12 y}$?

A: To evaluate the options, you can compare the final simplified expression with each of the options. You can rewrite the expression as follows:

$= 10 \cdot x^{7/2} \cdot y^{3/2} \cdot \sqrt{6}$

This expression matches option B, which is $10 x^5 y \sqrt{6 x y}$. Therefore, the correct answer is option B.