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 . \[ D. \[ D . \[ 60 \sqrt{3} - 21

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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. 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 product of two expressions, specifically the product of a square root expression and a polynomial expression.

Understanding the Product of Expressions

The product of two expressions is obtained by multiplying each term of the first expression by each term of the second expression. For example, if we have two expressions:

a(x+y)a(x + y)

and

b(x−y)b(x - y)

then their product is:

ab(x+y)(x−y)ab(x + y)(x - y)

Simplifying the Product of Expressions

To simplify the product of expressions, we need to apply the distributive property, which states that for any three expressions a, b, and c:

a(b+c)=ab+aca(b + c) = ab + ac

Using this property, we can simplify the product of expressions by multiplying each term of the first expression by each term of the second expression.

Simplifying the Given Product

Now, let's simplify the given product:

32(56−73)3 \sqrt{2}(5 \sqrt{6} - 7 \sqrt{3})

Using the distributive property, we can simplify this product as follows:

32(56−73)=32⋅56−32⋅733 \sqrt{2}(5 \sqrt{6} - 7 \sqrt{3}) = 3 \sqrt{2} \cdot 5 \sqrt{6} - 3 \sqrt{2} \cdot 7 \sqrt{3}

=1512−216= 15 \sqrt{12} - 21 \sqrt{6}

Now, we can simplify the square root expressions:

12=4⋅3=23\sqrt{12} = \sqrt{4 \cdot 3} = 2 \sqrt{3}

6=2⋅3=2⋅3\sqrt{6} = \sqrt{2 \cdot 3} = \sqrt{2} \cdot \sqrt{3}

Substituting these values, we get:

1512−216=15⋅23−212⋅315 \sqrt{12} - 21 \sqrt{6} = 15 \cdot 2 \sqrt{3} - 21 \sqrt{2} \cdot \sqrt{3}

=303−216= 30 \sqrt{3} - 21 \sqrt{6}

Comparing the Simplified Product with the Options

Now, let's compare the simplified product with the options:

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 simplified product matches option C.

Conclusion

In this article, we simplified the product of two expressions using the distributive property. We also compared the simplified product with the options and found that it matches option C. This exercise demonstrates the importance of simplifying products of expressions in mathematics.

Key Takeaways

  • The product of two expressions is obtained by multiplying each term of the first expression by each term of the second expression.
  • The distributive property states that for any three expressions a, b, and c: a(b + c) = ab + ac.
  • To simplify the product of expressions, we need to apply the distributive property.
  • The simplified product of two expressions can be compared with the options to determine the correct answer.

Final Answer

The final answer is:

In this article, we will address some of the most frequently asked questions related to the product of expressions, specifically the product of a square root expression and a polynomial expression.

Q: What is the product of two expressions?

A: The product of two expressions is obtained by multiplying each term of the first expression by each term of the second expression.

Q: How do I simplify the product of expressions?

A: To simplify the product of expressions, you need to apply the distributive property, which states that for any three expressions a, b, and c: a(b + c) = ab + ac.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in mathematics that states that for any three expressions a, b, and c: a(b + c) = ab + ac.

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

A: To apply the distributive property, you need to multiply each term of the first expression by each term of the second expression.

Q: What are some common mistakes to avoid when simplifying the product of expressions?

A: Some common mistakes to avoid when simplifying the product of expressions include:

  • Not applying the distributive property correctly
  • Not simplifying the square root expressions
  • Not combining like terms

Q: How do I simplify square root expressions?

A: To simplify square root expressions, you need to look for perfect squares that can be factored out of the expression.

Q: What is a perfect square?

A: A perfect square is a number that can be expressed as the square of an integer, such as 4, 9, 16, etc.

Q: How do I factor out perfect squares from square root expressions?

A: To factor out perfect squares from square root expressions, you need to look for the largest perfect square that can be factored out of the expression.

Q: What are some examples of simplifying the product of expressions?

A: Here are some examples of simplifying the product of expressions:

  • Simplifying the product of two binomials: (x + y)(x - y) = x^2 - y^2
  • Simplifying the product of a square root expression and a polynomial expression: 3√2(5√6 - 7√3) = 30√3 - 21√6

Q: How do I determine the correct answer when simplifying the product of expressions?

A: To determine the correct answer when simplifying the product of expressions, you need to compare the simplified product with the options and choose the one that matches.

Conclusion

In this article, we addressed some of the most frequently asked questions related to the product of expressions, specifically the product of a square root expression and a polynomial expression. We also provided some examples of simplifying the product of expressions and tips for determining the correct answer.

Key Takeaways

  • The product of two expressions is obtained by multiplying each term of the first expression by each term of the second expression.
  • The distributive property states that for any three expressions a, b, and c: a(b + c) = ab + ac.
  • To simplify the product of expressions, you need to apply the distributive property and simplify the square root expressions.
  • To determine the correct answer when simplifying the product of expressions, you need to compare the simplified product with the options and choose the one that matches.

Final Answer

The final answer is:

C. 303−21630 \sqrt{3} - 21 \sqrt{6}