Understanding the Product Notation

Oct 13, 2025 by recoveryshake.com 141 views

In mathematics, the product of two or more expressions is a fundamental concept that is used extensively in various branches of mathematics, including algebra, geometry, and calculus. When we are given a product of expressions, we need to simplify it to a single expression. In this article, we will explore how to simplify the given product and find the correct answer.

Understanding the Product Notation

The given product is represented as $3 \sqrt{2}(5 \sqrt{6} - 7 \sqrt{3})$ . This notation indicates that we need to multiply the expression $3 \sqrt{2}$ by the expression $(5 \sqrt{6} - 7 \sqrt{3})$ . To simplify this product, we need to apply the distributive property, which states that for any real numbers $a$ , $b$ , and $c$ , we have $a(b + c) = ab + ac$ .

Applying the Distributive Property

To simplify the given product, we will apply the distributive property. We will multiply the expression $3 \sqrt{2}$ by each term in the expression $(5 \sqrt{6} - 7 \sqrt{3})$ . This will give us two separate products, which we will then combine to get the final result.

Step 1: Multiply $3 \sqrt{2}$ by $5 \sqrt{6}$

To multiply $3 \sqrt{2}$ by $5 \sqrt{6}$ , we need to multiply the coefficients and the radicals separately. The coefficient of $3 \sqrt{2}$ is $3$ , and the coefficient of $5 \sqrt{6}$ is $5$ . When we multiply these coefficients, we get $3 \times 5 = 15$ . The radicals are $\sqrt{2}$ and $\sqrt{6}$ , which can be multiplied by multiplying their radicands, i.e., $2$ and $6$ . This gives us $\sqrt{2} \times \sqrt{6} = \sqrt{12}$ . However, we can simplify $\sqrt{12}$ to $2\sqrt{3}$ .

So, the product of $3 \sqrt{2}$ and $5 \sqrt{6}$ is $15 \times 2\sqrt{3} = 30\sqrt{3}$ .

Step 2: Multiply $3 \sqrt{2}$ by $-7 \sqrt{3}$

To multiply $3 \sqrt{2}$ by $-7 \sqrt{3}$ , we need to multiply the coefficients and the radicals separately. The coefficient of $3 \sqrt{2}$ is $3$ , and the coefficient of $-7 \sqrt{3}$ is $-7$ . When we multiply these coefficients, we get $3 \times -7 = -21$ . The radicals are $\sqrt{2}$ and $\sqrt{3}$ , which can be multiplied by multiplying their radicands, i.e., $2$ and $3$ . This gives us $\sqrt{2} \times \sqrt{3} = \sqrt{6}$ .

So, the product of $3 \sqrt{2}$ and $-7 \sqrt{3}$ is $-21 \times \sqrt{6} = -21\sqrt{6}$ .

Combining the Products

Now that we have simplified the two separate products, we can combine them to get the final result. We have $30\sqrt{3}$ and $-21\sqrt{6}$ , which can be combined by adding them together.

So, the final result is $30\sqrt{3} - 21\sqrt{6}$ .

Conclusion

In this article, we have explored how to simplify the given product and find the correct answer. We applied the distributive property to multiply the expression $3 \sqrt{2}$ by the expression $(5 \sqrt{6} - 7 \sqrt{3})$ . We then combined the two separate products to get the final result, which is $30\sqrt{3} - 21\sqrt{6}$ .

Answer

The correct answer is:

C. $30 \sqrt{3} - 21 \sqrt{6}$

In this article, we will address some of the most common questions related to the product of expressions and provide detailed answers to help you understand the concept better.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in mathematics that states that for any real numbers $a$ , $b$ , and $c$ , we have $a(b + c) = ab + ac$ . This property allows us to distribute a single term across the terms inside the parentheses.

Q: How do I apply the distributive property to simplify a product?

A: To apply the distributive property, you need to multiply the expression outside the parentheses by each term inside the parentheses. This will give you two separate products, which you can then combine to get the final result.

Q: What is the difference between multiplying radicals and multiplying coefficients?

A: When multiplying radicals, you need to multiply the radicands (the numbers inside the square root symbol) and keep the same radical sign. When multiplying coefficients, you simply multiply the numbers together.

Q: Can I simplify a product by combining like terms?

A: Yes, you can simplify a product by combining like terms. Like terms are terms that have the same variable and exponent. When you combine like terms, you add or subtract the coefficients of the like terms.

Q: How do I know which answer is correct?

A: To determine which answer is correct, you need to simplify the product and compare it to the answer choices. Make sure to follow the order of operations and apply the distributive property correctly.

Q: What if I get a negative answer?

A: If you get a negative answer, it means that the product is negative. This is because the distributive property allows us to distribute a negative term across the terms inside the parentheses.

Q: Can I use the distributive property to simplify a product with multiple parentheses?

A: Yes, you can use the distributive property to simplify a product with multiple parentheses. Simply apply the distributive property to each set of parentheses separately and then combine the results.

Q: How do I know when to use the distributive property?

A: You should use the distributive property whenever you need to simplify a product with multiple terms inside the parentheses. This will help you to distribute the single term across the terms inside the parentheses and simplify the product.

Q: Can I use the distributive property to simplify a product with variables and constants?

A: Yes, you can use the distributive property to simplify a product with variables and constants. Simply apply the distributive property to each term separately and then combine the results.

Conclusion

In this article, we have addressed some of the most common questions related to the product of expressions and provided detailed answers to help you understand the concept better. We have also discussed how to apply the distributive property to simplify a product and how to determine which answer is correct. By following the steps outlined in this article, you should be able to simplify products with ease and confidence.

Additional Resources

If you need additional help or resources, you can try the following:

Check out online tutorials and videos that demonstrate how to apply the distributive property.
Practice simplifying products with multiple terms inside the parentheses.
Use online calculators or software to check your work and ensure that you are getting the correct answer.

By following these tips and practicing regularly, you should be able to master the distributive property and simplify products with ease.