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}\$}

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Understanding the Problem

The given problem involves simplifying a mathematical expression that contains square roots. To solve this problem, 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 expression, 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 need to simplify each term separately. We can start by simplifying the first term, which is 32â‹…563 \sqrt{2} \cdot 5 \sqrt{6}.

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

Since 12\sqrt{12} can be simplified further, we can rewrite it as 4â‹…3\sqrt{4 \cdot 3}, which is equal to 232 \sqrt{3}.

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

Now, let's simplify the second term, which is 32â‹…733 \sqrt{2} \cdot 7 \sqrt{3}.

3 \sqrt{2} \cdot 7 \sqrt{3} = 21 \sqrt{6}

Step 3: Combine the Terms

Now that we have simplified both terms, we can combine them to get the final expression.

3 \sqrt{2}(5 \sqrt{6} - 7 \sqrt{3}) = 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 our final answer with the options provided.

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

Our final answer matches with Option C, which is 303−21630 \sqrt{3} - 21 \sqrt{6}.

Key Takeaways

  • To simplify a mathematical expression that contains square roots, we need to apply the rules of algebra and simplify the expression step by step.
  • We can distribute the square root to both terms inside the parentheses and then simplify each term separately.
  • We can combine the simplified terms to get the final expression.

Final Answer

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions related to the problem of simplifying a mathematical expression that contains square roots.

Q: What is the rule for simplifying square roots?

A: The rule for simplifying square roots is to look for perfect squares inside the square root. If we find a perfect square, we can simplify the square root by taking the square root of the perfect square.

Q: How do we distribute the square root to both terms inside the parentheses?

A: To distribute the square root to both terms inside the parentheses, we need to multiply the square root by each term separately. This will give us two separate terms that we can simplify separately.

Q: Can we simplify the square root of a product?

A: Yes, we can simplify the square root of a product by taking the square root of each factor separately. This will give us a simplified expression that contains the square roots of each factor.

Q: How do we simplify the square root of a difference?

A: To simplify the square root of a difference, we need to look for perfect squares inside the square root. If we find a perfect square, we can simplify the square root by taking the square root of the perfect square.

Q: Can we simplify the square root of a fraction?

A: Yes, we can simplify the square root of a fraction by taking the square root of the numerator and the denominator separately. This will give us a simplified expression that contains the square roots of the numerator and the denominator.

Q: How do we simplify the square root of a variable?

A: To simplify the square root of a variable, we need to look for perfect squares inside the square root. If we find a perfect square, we can simplify the square root by taking the square root of the perfect square.

Q: Can we simplify the square root of a negative number?

A: No, we cannot simplify the square root of a negative number. The square root of a negative number is an imaginary number, and it cannot be simplified further.

Q: How do we simplify a mathematical expression that contains square roots?

A: To simplify a mathematical expression that contains square roots, we need to apply the rules of algebra and simplify the expression step by step. We can distribute the square root to both terms inside the parentheses, simplify each term separately, and then combine the simplified terms to get the final expression.

Conclusion

In this article, we have answered some of the most frequently asked questions related to the problem of simplifying a mathematical expression that contains square roots. We have discussed the rules for simplifying square roots, distributing the square root to both terms inside the parentheses, simplifying the square root of a product, and simplifying the square root of a difference.

Key Takeaways

  • To simplify a mathematical expression that contains square roots, we need to apply the rules of algebra and simplify the expression step by step.
  • We can distribute the square root to both terms inside the parentheses and then simplify each term separately.
  • We can simplify the square root of a product by taking the square root of each factor separately.
  • We can simplify the square root of a difference by looking for perfect squares inside the square root.

Final Answer

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