Find The Product And Simplify Your Answer. ( 3 J − 3 ) ( 3 J + 3 (3j - 3)(3j + 3 ( 3 J − 3 ) ( 3 J + 3 ] □ \square □

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Introduction

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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. The product of two expressions is obtained by multiplying each term of the first expression by each term of the second expression. In this article, we will focus on finding the product and simplifying the expression $(3j - 3)(3j + 3)$.

Understanding the Concept of Product

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The product of two expressions is a way of combining them to obtain a new expression. When we multiply two expressions, we are essentially combining their terms to obtain a new expression. For example, if we have two expressions $x$ and $y$, their product is obtained by multiplying each term of $x$ by each term of $y$. This can be represented as:

$$xy = (x_1y_1 + x_1y_2 + ... + x_1y_n) + (x_2y_1 + x_2y_2 + ... + x_2y_n) + ... + (x_ny_1 + x_ny_2 + ... + x_ny_n)$$

where $x_i$ and $y_i$ are the terms of the expressions $x$ and $y$, respectively.

Finding the Product of $(3j - 3)$ and $(3j + 3)$

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To find the product of $(3j - 3)$ and $(3j + 3)$, we need to multiply each term of the first expression by each term of the second expression. This can be done using the distributive property of multiplication over addition.

$$(3j - 3)(3j + 3) = (3j)(3j) + (3j)(3) - (3)(3j) - (3)(3)$$

Using the commutative and associative properties of multiplication, we can simplify the expression as:

$$(3j)(3j) + (3j)(3) - (3)(3j) - (3)(3) = 9j^2 + 9j - 9j - 9$$

Simplifying the Expression

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Now that we have found the product of $(3j - 3)$ and $(3j + 3)$, we can simplify the expression by combining like terms. In this case, we have two terms that are opposites of each other, $9j$ and $-9j$. These terms can be combined to obtain:

$$9j^2 + 9j - 9j - 9 = 9j^2 - 9$$

Conclusion

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In this article, we have found the product and simplified the expression $(3j - 3)(3j + 3)$. We have used the distributive property of multiplication over addition to find the product, and then simplified the expression by combining like terms. The final simplified expression is $9j^2 - 9$. This result can be used in various mathematical applications, including algebra, geometry, and calculus.

Frequently Asked Questions

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Q: What is the product of two expressions?

A: The product of two expressions is a way of combining them to obtain a new expression. When we multiply two expressions, we are essentially combining their terms to obtain a new expression.

Q: How do we find the product of two expressions?

A: To find the product of two expressions, we need to multiply each term of the first expression by each term of the second expression. This can be done using the distributive property of multiplication over addition.

Q: How do we simplify an expression?

A: To simplify an expression, we need to combine like terms. Like terms are terms that have the same variable and exponent. We can combine like terms by adding or subtracting their coefficients.

Final Answer

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The final answer is: $\boxed{9j^2 - 9}$

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Introduction

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In our previous article, we discussed how to find the product and simplify the expression $(3j - 3)(3j + 3)$. We used the distributive property of multiplication over addition to find the product, and then simplified the expression by combining like terms. In this article, we will provide a Q&A section to help you better understand the concept of finding the product and simplifying expressions.

Q&A

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Q: What is the difference between the product and the sum of two expressions?

A: The product of two expressions is a way of combining them to obtain a new expression by multiplying each term of the first expression by each term of the second expression. On the other hand, the sum of two expressions is a way of combining them to obtain a new expression by adding each term of the first expression to each term of the second expression.

Q: How do I know when to use the distributive property of multiplication over addition?

A: You should use the distributive property of multiplication over addition when you need to multiply two expressions. This property allows you to multiply each term of the first expression by each term of the second expression.

Q: Can I simplify an expression by combining like terms?

A: Yes, you can simplify an expression by combining like terms. Like terms are terms that have the same variable and exponent. You can combine like terms by adding or subtracting their coefficients.

Q: What is the final simplified expression for $(3j - 3)(3j + 3)$?

A: The final simplified expression for $(3j - 3)(3j + 3)$ is $9j^2 - 9$.

Q: Can I use the distributive property of multiplication over addition to find the product of three or more expressions?

A: Yes, you can use the distributive property of multiplication over addition to find the product of three or more expressions. However, you need to be careful when multiplying multiple expressions, as the number of terms in the product can become very large.

Q: How do I know when to use the commutative and associative properties of multiplication?

A: You should use the commutative and associative properties of multiplication when you need to rearrange the terms in an expression. These properties allow you to rearrange the terms in an expression without changing the value of the expression.

Q: Can I simplify an expression by canceling out like terms?

A: Yes, you can simplify an expression by canceling out like terms. However, you need to be careful when canceling out like terms, as this can change the value of the expression.

Tips and Tricks

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Tip 1: Use the distributive property of multiplication over addition to find the product of two expressions.

When multiplying two expressions, use the distributive property of multiplication over addition to find the product. This will help you to avoid making mistakes and ensure that you get the correct product.

Tip 2: Simplify expressions by combining like terms.

When simplifying an expression, combine like terms to get the final simplified expression. This will help you to avoid making mistakes and ensure that you get the correct simplified expression.

Tip 3: Use the commutative and associative properties of multiplication to rearrange terms.

When rearranging terms in an expression, use the commutative and associative properties of multiplication to ensure that you get the correct rearrangement.

Conclusion

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In this article, we have provided a Q&A section to help you better understand the concept of finding the product and simplifying expressions. We have also provided some tips and tricks to help you to find the product and simplify expressions correctly. By following these tips and tricks, you will be able to find the product and simplify expressions with ease.

Final Answer

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The final answer is: $\boxed{9j^2 - 9}$