What Is The Following Product?$\[3 \sqrt{2}(5 \sqrt{6}-7 \sqrt{3})\\]A. \[$30 \sqrt{2}-21 \sqrt{5}\$\]B. \[$60 \sqrt{2}-21 \sqrt{5}\$\]C. \[$30 \sqrt{3}-21 \sqrt{6}\$\]D. \[$60 \sqrt{3}-21 \sqrt{6}\$\]

by ADMIN 202 views

Understanding the Problem

The given problem involves simplifying a mathematical expression that includes square roots. To solve this, we need to apply the rules of algebra and simplify the expression step by step.

Step 1: Distribute the Square Root

The given expression is 32(56−73)3 \sqrt{2}(5 \sqrt{6}-7 \sqrt{3}). To simplify this, we need to distribute the square root of 2 to both terms inside the parentheses.

3 \sqrt{2}(5 \sqrt{6}-7 \sqrt{3}) = 3 \sqrt{2} \cdot 5 \sqrt{6} - 3 \sqrt{2} \cdot 7 \sqrt{3}

Step 2: Simplify the Terms

Now, we can simplify each term separately. We know that 2⋅6=12\sqrt{2} \cdot \sqrt{6} = \sqrt{12} and 2⋅3=6\sqrt{2} \cdot \sqrt{3} = \sqrt{6}.

3 \sqrt{2} \cdot 5 \sqrt{6} = 15 \sqrt{12}
3 \sqrt{2} \cdot 7 \sqrt{3} = 21 \sqrt{6}

Step 3: Simplify the Square Roots

We can simplify the square roots further by expressing them in terms of their prime factors. We know that 12=4⋅3=23\sqrt{12} = \sqrt{4 \cdot 3} = 2 \sqrt{3}.

15 \sqrt{12} = 15 \cdot 2 \sqrt{3} = 30 \sqrt{3}
21 \sqrt{6} = 21 \sqrt{2} \cdot \sqrt{3} = 21 \sqrt{6}

Step 4: Combine the Terms

Now, we can combine the simplified terms to get the final expression.

30 \sqrt{3} - 21 \sqrt{6}

Conclusion

The final answer is 303−21630 \sqrt{3} - 21 \sqrt{6}.

Comparison with the Options

Let's compare the final answer with the options given:

A. 302−21530 \sqrt{2}-21 \sqrt{5} B. 602−21560 \sqrt{2}-21 \sqrt{5} C. 303−21630 \sqrt{3}-21 \sqrt{6} D. 603−21660 \sqrt{3}-21 \sqrt{6}

The final answer matches option C.

Final Answer

Understanding the Problem

The given problem involves simplifying a mathematical expression that includes square roots. To solve this, we need to apply the rules of algebra and simplify the expression step by step.

Q: What is the given expression?

A: The given expression is 32(56−73)3 \sqrt{2}(5 \sqrt{6}-7 \sqrt{3}).

Q: How do we simplify the expression?

A: To simplify the expression, we need to distribute the square root of 2 to both terms inside the parentheses.

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

A: The first step is to distribute the square root of 2 to both terms inside the parentheses.

Q: How do we simplify the terms?

A: We can simplify each term separately. We know that 2⋅6=12\sqrt{2} \cdot \sqrt{6} = \sqrt{12} and 2⋅3=6\sqrt{2} \cdot \sqrt{3} = \sqrt{6}.

Q: Can we simplify the square roots further?

A: Yes, we can simplify the square roots further by expressing them in terms of their prime factors. We know that 12=4⋅3=23\sqrt{12} = \sqrt{4 \cdot 3} = 2 \sqrt{3}.

Q: What is the final expression after simplifying the terms?

A: The final expression is 303−21630 \sqrt{3} - 21 \sqrt{6}.

Q: How does the final answer compare with the options given?

A: The final answer matches option C.

Q: What is the final answer?

A: The final answer is C.303−216\boxed{C. 30 \sqrt{3}-21 \sqrt{6}}.

Frequently Asked Questions (FAQs)

Q: What is the rule for simplifying expressions with square roots?

A: The rule is to distribute the square root to both terms inside the parentheses and then simplify each term separately.

Q: How do we simplify expressions with multiple square roots?

A: We can simplify expressions with multiple square roots by expressing them in terms of their prime factors and then simplifying each term separately.

Q: What is the importance of simplifying expressions with square roots?

A: Simplifying expressions with square roots is important because it helps us to evaluate the expression more easily and accurately.

Q: Can we use a calculator to simplify expressions with square roots?

A: Yes, we can use a calculator to simplify expressions with square roots. However, it's always a good idea to check the answer by simplifying the expression manually.

Conclusion

Simplifying expressions with square roots is an important skill in mathematics. By following the rules and steps outlined in this article, you can simplify expressions with square roots and evaluate them more easily and accurately.