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 $x \geq 0$ and $y \geq 0$, which means both $x$ and $y$ are non-negative numbers. The expression we need to simplify is $\sqrt{5 x^8 y^2} \cdot \sqrt{10 x^3} \cdot \sqrt{12 y}$.

Breaking Down the Expression

To simplify the given expression, we need to apply the properties of square roots. We know that the square root of a product is equal to the product of the square roots. Therefore, we can rewrite the expression as:

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

Simplifying the Square Roots

Now, we can simplify each square root individually. We know that the square root of a number raised to an even power is equal to the number raised to half of that power. Therefore, we can rewrite each square root as:

$\sqrt{5} = \sqrt{5}$ $\sqrt{x^8} = x^4$ $\sqrt{y^2} = y$ $\sqrt{10} = \sqrt{10}$ $\sqrt{x^3} = x^{3/2}$ $\sqrt{12} = \sqrt{12}$ $\sqrt{y} = \sqrt{y}$

Combining the Simplified Square Roots

Now, we can combine the simplified square roots to get the final expression:

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

Simplifying the Expression Further

We can simplify the expression further by combining like terms. We know that the product of two numbers raised to a power is equal to the product of the numbers raised to that power. Therefore, we can rewrite the expression as:

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

Simplifying the Exponents

Now, we can simplify the exponents by combining like terms. We know that the sum of two numbers raised to a power is equal to the product of the numbers raised to that power. Therefore, we can rewrite the expression as:

$\sqrt{5} \cdot \sqrt{10} \cdot x^{4+3/2} \cdot y \cdot \sqrt{12} \cdot \sqrt{y} = \sqrt{5} \cdot \sqrt{10} \cdot x^{7/2} \cdot y \cdot \sqrt{12} \cdot \sqrt{y}$

Simplifying the Square Roots of Numbers

Now, we can simplify the square roots of numbers by rewriting them as the product of the number and the square root of the number. Therefore, we can rewrite the expression as:

$\sqrt{5} \cdot \sqrt{10} \cdot x^{7/2} \cdot y \cdot \sqrt{12} \cdot \sqrt{y} = \sqrt{5} \cdot \sqrt{10} \cdot x^{7/2} \cdot y \cdot \sqrt{12} \cdot y^{1/2}$

Simplifying the Expression Further

We can simplify the expression further by combining like terms. We know that the product of two numbers raised to a power is equal to the product of the numbers raised to that power. Therefore, we can rewrite the expression as:

$\sqrt{5} \cdot \sqrt{10} \cdot x^{7/2} \cdot y \cdot \sqrt{12} \cdot y^{1/2} = \sqrt{5} \cdot \sqrt{10} \cdot x^{7/2} \cdot y^{3/2} \cdot \sqrt{12}$

Simplifying the Square Roots of Numbers

Now, we can simplify the square roots of numbers by rewriting them as the product of the number and the square root of the number. Therefore, we can rewrite the expression as:

$\sqrt{5} \cdot \sqrt{10} \cdot x^{7/2} \cdot y^{3/2} \cdot \sqrt{12} = \sqrt{5} \cdot \sqrt{10} \cdot x^{7/2} \cdot y^{3/2} \cdot \sqrt{4 \cdot 3}$

Simplifying the Expression Further

We can simplify the expression further by combining like terms. We know that the product of two numbers raised to a power is equal to the product of the numbers raised to that power. Therefore, we can rewrite the expression as:

$\sqrt{5} \cdot \sqrt{10} \cdot x^{7/2} \cdot y^{3/2} \cdot \sqrt{4 \cdot 3} = \sqrt{5} \cdot \sqrt{10} \cdot x^{7/2} \cdot y^{3/2} \cdot 2 \cdot \sqrt{3}$

Simplifying the Square Roots of Numbers

Now, we can simplify the square roots of numbers by rewriting them as the product of the number and the square root of the number. Therefore, we can rewrite the expression as:

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

Simplifying the Expression Further

We can simplify the expression further by combining like terms. We know that the product of two numbers raised to a power is equal to the product of the numbers raised to that power. Therefore, we can rewrite the expression as:

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

Final Answer

After simplifying the expression, we get:

$\sqrt{5} \cdot \sqrt{10} \cdot x^{7/2} \cdot y^{3/2} \cdot 2 \cdot \sqrt{3} = 10 x^{7/2} y^{3/2} \sqrt{6}$

Conclusion

The final answer is $10 x^{7/2} y^{3/2} \sqrt{6}$. This is the simplified form of the given expression.

Discussion

The given problem involves simplifying a mathematical expression that includes square roots and variables. We assumed that $x \geq 0$ and $y \geq 0$, which means both $x$ and $y$ are non-negative numbers. We applied the properties of square roots to simplify the expression and arrived at the final answer.

Final Answer Comparison

We can compare the final answer with the given options to determine which one is correct.

• Option A: $3 x^5 y \sqrt{3 x y}$
• Option B: $10 x^5 y \sqrt{6 x y}$
• Option C: $3 x^3 y \sqrt{3 x^2 y}$

The final answer $10 x^{7/2} y^{3/2} \sqrt{6}$ matches with option B: $10 x^5 y \sqrt{6 x y}$. Therefore, the correct answer is option B.

Understanding the Problem

The given problem involves simplifying a mathematical expression that includes square roots and variables. We are asked to assume that $x \geq 0$ and $y \geq 0$, which means both $x$ and $y$ are non-negative numbers. The expression we need to simplify is $\sqrt{5 x^8 y^2} \cdot \sqrt{10 x^3} \cdot \sqrt{12 y}$.

Q&A

Q: What is the first step in simplifying the given expression?

A: The first step in simplifying the given expression is to apply the properties of square roots. We know that the square root of a product is equal to the product of the square roots. Therefore, we can rewrite the expression as:

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

Q: How do we simplify the square roots of numbers?

A: We can simplify the square roots of numbers by rewriting them as the product of the number and the square root of the number. Therefore, we can rewrite the expression as:

$\sqrt{5} = \sqrt{5}$ $\sqrt{x^8} = x^4$ $\sqrt{y^2} = y$ $\sqrt{10} = \sqrt{10}$ $\sqrt{x^3} = x^{3/2}$ $\sqrt{12} = \sqrt{12}$ $\sqrt{y} = \sqrt{y}$

Q: How do we combine the simplified square roots?

A: We can combine the simplified square roots to get the final expression:

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

Q: How do we simplify the expression further?

A: We can simplify the expression further by combining like terms. We know that the product of two numbers raised to a power is equal to the product of the numbers raised to that power. Therefore, we can rewrite the expression as:

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

Q: What is the final answer?

A: After simplifying the expression, we get:

$\sqrt{5} \cdot \sqrt{10} \cdot x^{4+3/2} \cdot y \cdot \sqrt{12} \cdot \sqrt{y} = 10 x^{7/2} y^{3/2} \sqrt{6}$

Conclusion

The final answer is $10 x^{7/2} y^{3/2} \sqrt{6}$. This is the simplified form of the given expression.

Discussion

The given problem involves simplifying a mathematical expression that includes square roots and variables. We assumed that $x \geq 0$ and $y \geq 0$, which means both $x$ and $y$ are non-negative numbers. We applied the properties of square roots to simplify the expression and arrived at the final answer.

Final Answer Comparison

We can compare the final answer with the given options to determine which one is correct.

• Option A: $3 x^5 y \sqrt{3 x y}$
• Option B: $10 x^5 y \sqrt{6 x y}$
• Option C: $3 x^3 y \sqrt{3 x^2 y}$

The final answer $10 x^{7/2} y^{3/2} \sqrt{6}$ matches with option B: $10 x^5 y \sqrt{6 x y}$. Therefore, the correct answer is option B.

Tips and Tricks

• When simplifying expressions with square roots, it's essential to apply the properties of square roots.
• When combining like terms, make sure to add the exponents of the same base.
• When simplifying expressions with variables, make sure to use the correct exponent rules.

Common Mistakes

• Failing to apply the properties of square roots.
• Failing to combine like terms correctly.
• Failing to use the correct exponent rules when simplifying expressions with variables.

Conclusion

Simplifying expressions with square roots and variables can be challenging, but with practice and patience, you can master the skills. Remember to apply the properties of square roots, combine like terms correctly, and use the correct exponent rules when simplifying expressions with variables.