If $A=15 \sqrt{48}$ And $B=6 \sqrt{150}$, Then $\frac{A}{B}$ Is Equal To:A. $\frac{18 \sqrt{2}}{5}$B. $\sqrt{2}$C. $\frac{19 \sqrt{2}}{42}$D. $2 \sqrt{2}$

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Understanding the Problem

To find the value of $\frac{A}{B}$, we need to first simplify the expressions for $A$ and $B$. The given expressions are $A=15 \sqrt{48}$ and $B=6 \sqrt{150}$. We can simplify these expressions by factoring the numbers inside the square roots.

Simplifying the Expressions

We can start by simplifying the expression for $A$. The expression $A=15 \sqrt{48}$ can be simplified as follows:

$$A=15 \sqrt{48} = 15 \sqrt{16 \times 3} = 15 \times 4 \sqrt{3} = 60 \sqrt{3}$$

Similarly, we can simplify the expression for $B$. The expression $B=6 \sqrt{150}$ can be simplified as follows:

$$B=6 \sqrt{150} = 6 \sqrt{25 \times 6} = 6 \times 5 \sqrt{6} = 30 \sqrt{6}$$

Simplifying the Square Roots

Now that we have simplified the expressions for $A$ and $B$, we can simplify the square roots. We can rewrite $\sqrt{3}$ as $\sqrt{3} = \sqrt{3 \times 2} = \sqrt{6} \times \frac{\sqrt{3}}{\sqrt{6}} = \sqrt{2} \times \sqrt{3}$.

Similarly, we can rewrite $\sqrt{6}$ as $\sqrt{6} = \sqrt{6 \times 2} = \sqrt{12} \times \frac{\sqrt{6}}{\sqrt{12}} = \sqrt{2} \times \sqrt{6}$.

Simplifying the Expression for A

Now that we have simplified the square roots, we can simplify the expression for $A$. We can rewrite $A=60 \sqrt{3}$ as follows:

$$A=60 \sqrt{3} = 60 \times \sqrt{3 \times 2} = 60 \times \sqrt{2} \times \sqrt{3} = 60 \sqrt{2} \times \sqrt{3}$$

Simplifying the Expression for B

Similarly, we can simplify the expression for $B$. We can rewrite $B=30 \sqrt{6}$ as follows:

$$B=30 \sqrt{6} = 30 \times \sqrt{6 \times 2} = 30 \times \sqrt{2} \times \sqrt{6} = 30 \sqrt{2} \times \sqrt{6}$$

Finding the Value of $\frac{A}{B}$

Now that we have simplified the expressions for $A$ and $B$, we can find the value of $\frac{A}{B}$. We can rewrite $\frac{A}{B}$ as follows:

$$\frac{A}{B} = \frac{60 \sqrt{2} \times \sqrt{3}}{30 \sqrt{2} \times \sqrt{6}}$$

Canceling Out the Common Factors

We can cancel out the common factors in the numerator and denominator. We can rewrite $\frac{A}{B}$ as follows:

$$\frac{A}{B} = \frac{60 \sqrt{2} \times \sqrt{3}}{30 \sqrt{2} \times \sqrt{6}} = \frac{2 \times 30 \sqrt{2} \times \sqrt{3}}{30 \sqrt{2} \times \sqrt{6}}$$

Canceling Out the Common Factors

We can cancel out the common factors in the numerator and denominator. We can rewrite $\frac{A}{B}$ as follows:

$$\frac{A}{B} = \frac{2 \times 30 \sqrt{2} \times \sqrt{3}}{30 \sqrt{2} \times \sqrt{6}} = \frac{2 \times \sqrt{3}}{\sqrt{6}}$$

Simplifying the Expression

We can simplify the expression by rewriting $\sqrt{6}$ as $\sqrt{2} \times \sqrt{3}$. We can rewrite $\frac{A}{B}$ as follows:

$$\frac{A}{B} = \frac{2 \times \sqrt{3}}{\sqrt{2} \times \sqrt{3}} = \frac{2}{\sqrt{2}}$$

Rationalizing the Denominator

We can rationalize the denominator by multiplying the numerator and denominator by $\sqrt{2}$. We can rewrite $\frac{A}{B}$ as follows:

$$\frac{A}{B} = \frac{2}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{2 \sqrt{2}}{2} = \sqrt{2}$$

Conclusion

Therefore, the value of $\frac{A}{B}$ is $\boxed{\sqrt{2}}$.

Final Answer

The final answer is $\boxed{\sqrt{2}}$.

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Q: What is the value of $A=15 \sqrt{48}$?

A: The value of $A=15 \sqrt{48}$ can be simplified as follows:

$$A=15 \sqrt{48} = 15 \sqrt{16 \times 3} = 15 \times 4 \sqrt{3} = 60 \sqrt{3}$$

Q: What is the value of $B=6 \sqrt{150}$?

A: The value of $B=6 \sqrt{150}$ can be simplified as follows:

$$B=6 \sqrt{150} = 6 \sqrt{25 \times 6} = 6 \times 5 \sqrt{6} = 30 \sqrt{6}$$

Q: How do we simplify the square roots in the expressions for $A$ and $B$?

A: We can simplify the square roots by rewriting them as follows:

$$\sqrt{3} = \sqrt{3 \times 2} = \sqrt{6} \times \frac{\sqrt{3}}{\sqrt{6}} = \sqrt{2} \times \sqrt{3}$$

$$\sqrt{6} = \sqrt{6 \times 2} = \sqrt{12} \times \frac{\sqrt{6}}{\sqrt{12}} = \sqrt{2} \times \sqrt{6}$$

Q: How do we simplify the expression for $A$?

A: We can simplify the expression for $A$ by rewriting it as follows:

$$A=60 \sqrt{3} = 60 \times \sqrt{3 \times 2} = 60 \times \sqrt{2} \times \sqrt{3} = 60 \sqrt{2} \times \sqrt{3}$$

Q: How do we simplify the expression for $B$?

A: We can simplify the expression for $B$ by rewriting it as follows:

$$B=30 \sqrt{6} = 30 \times \sqrt{6 \times 2} = 30 \times \sqrt{2} \times \sqrt{6} = 30 \sqrt{2} \times \sqrt{6}$$

Q: How do we find the value of $\frac{A}{B}$?

A: We can find the value of $\frac{A}{B}$ by rewriting it as follows:

$$\frac{A}{B} = \frac{60 \sqrt{2} \times \sqrt{3}}{30 \sqrt{2} \times \sqrt{6}}$$

Q: How do we cancel out the common factors in the numerator and denominator?

A: We can cancel out the common factors by rewriting the expression as follows:

$$\frac{A}{B} = \frac{2 \times 30 \sqrt{2} \times \sqrt{3}}{30 \sqrt{2} \times \sqrt{6}} = \frac{2 \times \sqrt{3}}{\sqrt{6}}$$

Q: How do we simplify the expression further?

A: We can simplify the expression further by rewriting $\sqrt{6}$ as $\sqrt{2} \times \sqrt{3}$. We can rewrite the expression as follows:

$$\frac{A}{B} = \frac{2 \times \sqrt{3}}{\sqrt{2} \times \sqrt{3}} = \frac{2}{\sqrt{2}}$$

Q: How do we rationalize the denominator?

A: We can rationalize the denominator by multiplying the numerator and denominator by $\sqrt{2}$. We can rewrite the expression as follows:

$$\frac{A}{B} = \frac{2}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{2 \sqrt{2}}{2} = \sqrt{2}$$

Q: What is the final value of $\frac{A}{B}$?

A: The final value of $\frac{A}{B}$ is $\boxed{\sqrt{2}}$.

Q: What is the final answer?

A: The final answer is $\boxed{\sqrt{2}}$.