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

by ADMIN 209 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 multiplication and simplification of radicals. The expression to be simplified is:

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

Step 1: Distribute the Terms

To simplify the given expression, we need to distribute the terms inside the parentheses. This means we multiply each term inside the parentheses by the term outside the parentheses.

32(56−73)=32⋅56−32⋅73{ 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 Radicals

Now, we simplify the radicals by multiplying the numbers inside the square roots.

32⋅56=1512{ 3 \sqrt{2} \cdot 5 \sqrt{6} = 15 \sqrt{12} }

32⋅73=216{ 3 \sqrt{2} \cdot 7 \sqrt{3} = 21 \sqrt{6} }

Step 3: Combine Like Terms

We can now combine the like terms by adding or subtracting the coefficients of the radicals.

1512−216{ 15 \sqrt{12} - 21 \sqrt{6} }

Step 4: Simplify the Radicals Further

We can simplify the radicals further by expressing them in terms of their prime factors.

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 back into the expression, we get:

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

Step 5: Combine Like Terms Again

We can now combine the like terms again by adding or subtracting the coefficients of the radicals.

15⋅23−212⋅3=303−216{ 15 \cdot 2 \sqrt{3} - 21 \sqrt{2} \cdot \sqrt{3} = 30 \sqrt{3} - 21 \sqrt{6} }

Conclusion

The simplified expression is:

303−216{ 30 \sqrt{3} - 21 \sqrt{6} }

This matches option C.

Answer

Q: What is the product of two square roots?

A: The product of two square roots is the square root of the product of the numbers inside the square roots. For example, √2 × √3 = √(2 × 3) = √6.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to multiply the numbers inside the square roots and then simplify the resulting expression. You can also use the rules of multiplication and simplification of radicals to simplify the expression.

Q: What is the difference between a radical and a rational number?

A: A radical is an expression that contains a square root, such as √2 or √3. A rational number is a number that can be expressed as the ratio of two integers, such as 3/4 or 22/7.

Q: How do I add or subtract radicals?

A: To add or subtract radicals, you need to have the same index (or root) and the same radicand (or number inside the square root). For example, √2 + √2 = 2√2, but √2 + √3 is not equal to 2√5.

Q: What is the product of a radical and a rational number?

A: The product of a radical and a rational number is the product of the numbers inside the square root and the rational number. For example, √2 × 3 = 3√2.

Q: How do I simplify a radical expression with multiple terms?

A: To simplify a radical expression with multiple terms, you need to multiply the numbers inside the square roots and then simplify the resulting expression. You can also use the rules of multiplication and simplification of radicals to simplify the expression.

Q: What is the difference between a radical and an exponent?

A: A radical is an expression that contains a square root, such as √2 or √3. An exponent is a number that is raised to a power, such as 2^3 or 3^4.

Q: How do I simplify a radical expression with a negative exponent?

A: To simplify a radical expression with a negative exponent, you need to take the reciprocal of the number inside the square root and change the sign of the exponent. For example, √2^(-3) = 1/√2^3 = 1/2√2.

Q: What is the product of two radical expressions?

A: The product of two radical expressions is the product of the numbers inside the square roots. For example, √2 × √3 = √(2 × 3) = √6.

Q: How do I simplify a radical expression with a variable?

A: To simplify a radical expression with a variable, you need to multiply the numbers inside the square roots and then simplify the resulting expression. You can also use the rules of multiplication and simplification of radicals to simplify the expression.

Q: What is the difference between a radical and a polynomial?

A: A radical is an expression that contains a square root, such as √2 or √3. A polynomial is an expression that consists of variables and coefficients, such as 2x^2 + 3x - 4.

Q: How do I simplify a radical expression with a polynomial?

A: To simplify a radical expression with a polynomial, you need to multiply the numbers inside the square roots and then simplify the resulting expression. You can also use the rules of multiplication and simplification of radicals to simplify the expression.

Conclusion

Radicals are an important part of mathematics, and understanding how to simplify and manipulate them is crucial for solving problems in algebra, geometry, and other areas of mathematics. By following the rules of multiplication and simplification of radicals, you can simplify complex radical expressions and solve problems with ease.