Find The Product Of $8(4 \sqrt{7}-\sqrt{18})$.A. $32 \sqrt{7}-8 \sqrt{18}$B. $ 4 7 − 2 4 \sqrt{7}-\sqrt{2} 4 7 ​ − 2 ​ [/tex]C. $32 \sqrt{7}-24 \sqrt{2}$D. $4 \sqrt{7}-8 \sqrt{18}$

Understanding the Problem

To find the product of $8(4 \sqrt{7}-\sqrt{18})$, we need to apply the distributive property of multiplication over addition. This means that we will multiply each term inside the parentheses by the factor outside the parentheses, which is 8.

Applying the Distributive Property

The distributive property states that for any real numbers a, b, and c:

a(b + c) = ab + ac

Using this property, we can rewrite the given expression as:

$$8(4 \sqrt{7}-\sqrt{18}) = 8(4 \sqrt{7}) - 8(\sqrt{18})$$

Simplifying the Expression

Now, we can simplify each term separately.

Simplifying the First Term

The first term is $8(4 \sqrt{7})$. We can simplify this by multiplying 8 by 4, which gives us:

$$8(4 \sqrt{7}) = 32 \sqrt{7}$$

Simplifying the Second Term

The second term is $8(\sqrt{18})$. We can simplify this by multiplying 8 by the square root of 18. However, we need to simplify the square root of 18 first.

The square root of 18 can be simplified as:

$$\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3 \sqrt{2}$$

Now, we can multiply 8 by the simplified square root of 18:

$$8(\sqrt{18}) = 8(3 \sqrt{2}) = 24 \sqrt{2}$$

Combining the Terms

Now that we have simplified each term, we can combine them to get the final result:

$$8(4 \sqrt{7}-\sqrt{18}) = 32 \sqrt{7} - 24 \sqrt{2}$$

Conclusion

Therefore, the product of $8(4 \sqrt{7}-\sqrt{18})$ is $32 \sqrt{7} - 24 \sqrt{2}$.

Answer

The correct answer is:

C. $32 \sqrt{7}-24 \sqrt{2}$

Understanding the Problem

To find the product of $8(4 \sqrt{7}-\sqrt{18})$, we need to apply the distributive property of multiplication over addition. This means that we will multiply each term inside the parentheses by the factor outside the parentheses, which is 8.

Q&A

Q: What is the distributive property of multiplication over addition?

A: The distributive property states that for any real numbers a, b, and c:

a(b + c) = ab + ac

This means that we can multiply each term inside the parentheses by the factor outside the parentheses.

Q: How do we simplify the expression $8(4 \sqrt{7}-\sqrt{18})$?

A: We can simplify the expression by applying the distributive property and then simplifying each term separately.

Q: How do we simplify the first term $8(4 \sqrt{7})$?

A: We can simplify the first term by multiplying 8 by 4, which gives us:

$$8(4 \sqrt{7}) = 32 \sqrt{7}$$

Q: How do we simplify the second term $8(\sqrt{18})$?

A: We can simplify the second term by multiplying 8 by the square root of 18. However, we need to simplify the square root of 18 first.

The square root of 18 can be simplified as:

$$\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3 \sqrt{2}$$

Now, we can multiply 8 by the simplified square root of 18:

$$8(\sqrt{18}) = 8(3 \sqrt{2}) = 24 \sqrt{2}$$

Q: What is the final result of the expression $8(4 \sqrt{7}-\sqrt{18})$?

A: The final result of the expression is:

$$32 \sqrt{7} - 24 \sqrt{2}$$

Conclusion

Therefore, the product of $8(4 \sqrt{7}-\sqrt{18})$ is $32 \sqrt{7} - 24 \sqrt{2}$.

Answer

The correct answer is:

C. $32 \sqrt{7}-24 \sqrt{2}$

Additional Tips and Tricks

• When simplifying expressions, always look for opportunities to apply the distributive property.
• When simplifying square roots, look for perfect squares that can be factored out.
• When multiplying expressions, always multiply each term separately.

Common Mistakes to Avoid

• Failing to apply the distributive property when simplifying expressions.
• Not simplifying square roots when necessary.
• Not multiplying each term separately when multiplying expressions.

Conclusion

By following these tips and avoiding common mistakes, you can simplify expressions and find the product of $8(4 \sqrt{7}-\sqrt{18})$ with ease.