Simplify The Expression 3 × 12 \sqrt{3} \times \sqrt{12} 3 ​ × 12 ​ .A) 6 B) 2 3 2 \sqrt{3} 2 3 ​ C) 15 \sqrt{15} 15 ​ D) 3 2 3 \sqrt{2} 3 2 ​

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Understanding the Problem

When dealing with expressions involving square roots, it's essential to simplify them to make calculations easier. In this problem, we're given the expression $\sqrt{3} \times \sqrt{12}$, and we need to simplify it. To do this, we'll use the properties of square roots and multiplication.

Properties of Square Roots

Before we dive into simplifying the expression, let's recall some properties of square roots:

• The square root of a number is a value that, when multiplied by itself, gives the original number. For example, $\sqrt{16} = 4$ because $4 \times 4 = 16$.
• The square root of a product is equal to the product of the square roots. For example, $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$.
• The square root of a quotient is equal to the quotient of the square roots. For example, $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$.

Simplifying the Expression

Now that we've reviewed the properties of square roots, let's simplify the expression $\sqrt{3} \times \sqrt{12}$.

First, we can rewrite $\sqrt{12}$ as $\sqrt{4 \times 3}$ using the property of square roots mentioned earlier. This gives us:

$$\sqrt{3} \times \sqrt{12} = \sqrt{3} \times \sqrt{4 \times 3}$$

Next, we can use the property of square roots to simplify the expression further:

$$\sqrt{3} \times \sqrt{4 \times 3} = \sqrt{3 \times 4} \times \sqrt{3}$$

Now, we can simplify the expression inside the square root:

$$\sqrt{3 \times 4} = \sqrt{12}$$

So, we have:

$$\sqrt{3} \times \sqrt{12} = \sqrt{12} \times \sqrt{3}$$

Using the Commutative Property of Multiplication

The commutative property of multiplication states that the order of the factors does not change the product. In other words, $a \times b = b \times a$. We can use this property to rewrite the expression:

$$\sqrt{12} \times \sqrt{3} = \sqrt{3} \times \sqrt{12}$$

Simplifying the Expression Further

Now that we've used the commutative property of multiplication, we can simplify the expression further. We can rewrite $\sqrt{12}$ as $\sqrt{4 \times 3}$:

$$\sqrt{3} \times \sqrt{4 \times 3} = \sqrt{3} \times \sqrt{4} \times \sqrt{3}$$

Using the Property of Square Roots Again

We can use the property of square roots again to simplify the expression:

$$\sqrt{3} \times \sqrt{4} \times \sqrt{3} = \sqrt{3 \times 4} \times \sqrt{3}$$

Simplifying the Expression Inside the Square Root

We can simplify the expression inside the square root:

$$\sqrt{3 \times 4} = \sqrt{12}$$

So, we have:

$$\sqrt{3} \times \sqrt{12} = \sqrt{12} \times \sqrt{3}$$

Using the Property of Square Roots Again

We can use the property of square roots again to simplify the expression:

$$\sqrt{12} \times \sqrt{3} = \sqrt{3 \times 12}$$

Simplifying the Expression Inside the Square Root

We can simplify the expression inside the square root:

$$\sqrt{3 \times 12} = \sqrt{36}$$

Simplifying the Square Root

We can simplify the square root:

$$\sqrt{36} = 6$$

Conclusion

Therefore, the simplified expression is $\boxed{6}$.

Answer Choice

The correct answer is A) 6.

Discussion

This problem requires the use of the properties of square roots and multiplication. The key concept is to simplify the expression inside the square root and then use the commutative property of multiplication to rewrite the expression. The final answer is $\boxed{6}$.

Additional Tips

• When dealing with expressions involving square roots, it's essential to simplify them to make calculations easier.
• Use the properties of square roots to simplify the expression inside the square root.
• Use the commutative property of multiplication to rewrite the expression.
• Simplify the square root to get the final answer.

Related Problems

• Simplify the expression $\sqrt{2} \times \sqrt{8}$.
• Simplify the expression $\sqrt{5} \times \sqrt{20}$.
• Simplify the expression $\sqrt{7} \times \sqrt{49}$.

Practice Problems

• Simplify the expression $\sqrt{3} \times \sqrt{9}$.
• Simplify the expression $\sqrt{4} \times \sqrt{16}$.
• Simplify the expression $\sqrt{6} \times \sqrt{36}$.

Conclusion

In conclusion, simplifying the expression $\sqrt{3} \times \sqrt{12}$ requires the use of the properties of square roots and multiplication. The key concept is to simplify the expression inside the square root and then use the commutative property of multiplication to rewrite the expression. The final answer is $\boxed{6}$.

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Understanding the Problem

When dealing with expressions involving square roots, it's essential to simplify them to make calculations easier. In this problem, we're given the expression $\sqrt{3} \times \sqrt{12}$, and we need to simplify it. To do this, we'll use the properties of square roots and multiplication.

Q&A

Q: What is the property of square roots that we can use to simplify the expression?

A: The property of square roots that we can use to simplify the expression is $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$.

Q: How can we simplify the expression $\sqrt{12}$?

A: We can simplify the expression $\sqrt{12}$ by rewriting it as $\sqrt{4 \times 3}$.

Q: What is the commutative property of multiplication?

A: The commutative property of multiplication states that the order of the factors does not change the product. In other words, $a \times b = b \times a$.

Q: How can we use the commutative property of multiplication to rewrite the expression?

A: We can use the commutative property of multiplication to rewrite the expression as $\sqrt{3} \times \sqrt{12} = \sqrt{12} \times \sqrt{3}$.

Q: What is the final answer to the problem?

A: The final answer to the problem is $\boxed{6}$.

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

A: Some additional tips for simplifying expressions involving square roots include:

• Use the properties of square roots to simplify the expression inside the square root.
• Use the commutative property of multiplication to rewrite the expression.
• Simplify the square root to get the final answer.

Q: What are some related problems that we can try?

A: Some related problems that we can try include:

• Simplify the expression $\sqrt{2} \times \sqrt{8}$.
• Simplify the expression $\sqrt{5} \times \sqrt{20}$.
• Simplify the expression $\sqrt{7} \times \sqrt{49}$.

Q: What are some practice problems that we can try?

A: Some practice problems that we can try include:

• Simplify the expression $\sqrt{3} \times \sqrt{9}$.
• Simplify the expression $\sqrt{4} \times \sqrt{16}$.
• Simplify the expression $\sqrt{6} \times \sqrt{36}$.

Conclusion

In conclusion, simplifying the expression $\sqrt{3} \times \sqrt{12}$ requires the use of the properties of square roots and multiplication. The key concept is to simplify the expression inside the square root and then use the commutative property of multiplication to rewrite the expression. The final answer is $\boxed{6}$.