Using the Distributive Property

Oct 13, 2025 by miodeya.com 93 views

Simplify ${$ (2 \sqrt{5} - 4)(3 \sqrt{5} + 2)$}$

Understanding the Problem

When simplifying algebraic expressions, it's essential to apply the correct order of operations and use the distributive property to expand the expression. In this case, we're given the expression ${$ (2 \sqrt{5} - 4)(3 \sqrt{5} + 2)$}$ and need to simplify it.

Using the Distributive Property

To simplify the expression, we'll use the distributive property, which states that for any real numbers a, b, and c:

a(b + c) = ab + ac

We can apply this property to the given expression by multiplying each term in the first parentheses by each term in the second parentheses.

Step 1: Multiply the Terms

Using the distributive property, we multiply each term in the first parentheses by each term in the second parentheses:

${$ (2 \sqrt{5} - 4)(3 \sqrt{5} + 2)$ = (2 \sqrt{5})(3 \sqrt{5}) + (2 \sqrt{5})(2) + (-4)(3 \sqrt{5}) + (-4)(2)$

Step 2: Simplify the Terms

Now, we simplify each term:

[ $(2 \sqrt{5})(3 \sqrt{5}) = 6(5) = 30\$ \[$ (2 \sqrt{5})(2) = 4 \sqrt{5}$ [ $(-4)(3 \sqrt{5}) = -12 \sqrt{5}\$ \[$ (-4)(2) = -8$

Step 3: Combine Like Terms

Now, we combine like terms:

[$30 + 4 \sqrt{5} - 12 \sqrt{5} - 8$

Step 4: Simplify the Expression

Finally, we simplify the expression by combining like terms:

[$30 - 8 + 4 \sqrt{5} - 12 \sqrt{5}$ [$22 - 8 \sqrt{5}$

Conclusion

Therefore, the simplified expression is [$22 - 8 \sqrt{5}$.

Answer

The correct answer is:

D. [ $22 - 8 \sqrt{5}\$<br/> **Simplify \[$ (2 \sqrt{5} - 4)(3 \sqrt{5} + 2)$}$ Q&A**

Frequently Asked Questions

We've already simplified the expression [$(2 \sqrt{5} - 4)(3 \sqrt{5} + 2)$ in the previous section. However, we understand that you may still have some questions or doubts. Here are some frequently asked questions and answers to help you better understand the concept.

Q: What is the distributive property?

A: The distributive property is a mathematical concept that allows us to expand an expression by multiplying each term in one set of parentheses by each term in another set of parentheses.

Q: How do I apply the distributive property?

A: To apply the distributive property, simply multiply each term in the first set of parentheses by each term in the second set of parentheses. For example, in the expression [$(2 \sqrt{5} - 4)(3 \sqrt{5} + 2)$, we multiply each term in the first parentheses by each term in the second parentheses.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when simplifying an expression. The order of operations is:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify expressions with square roots?

A: To simplify expressions with square roots, we can use the following properties:

The square root of a product is equal to the product of the square roots: [$\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$.
The square root of a quotient is equal to the quotient of the square roots: [$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$.

Q: What is the difference between a rational expression and an irrational expression?

A: A rational expression is an expression that can be written as a fraction of two polynomials, such as [ $\frac{2x + 3}{x - 1}\$. An irrational expression, on the other hand, is an expression that cannot be written as a fraction of two polynomials, such as \[$ \sqrt{2x + 3}$.

Q: How do I simplify expressions with multiple terms?

A: To simplify expressions with multiple terms, we can use the following properties:

The sum of two or more terms is equal to the sum of the individual terms.
The product of two or more terms is equal to the product of the individual terms.

Conclusion

We hope this Q&A article has helped you better understand the concept of simplifying expressions with square roots. Remember to always apply the distributive property, follow the order of operations, and use the properties of square roots to simplify expressions. If you have any more questions or doubts, feel free to ask!

Additional Resources

If you're looking for more practice problems or want to learn more about simplifying expressions with square roots, here are some additional resources:

Khan Academy: Simplifying Expressions with Square Roots
Mathway: Simplifying Expressions with Square Roots
IXL: Simplifying Expressions with Square Roots

Practice Problems

Try simplifying the following expressions:

[$(3 \sqrt{2} + 2)(2 \sqrt{2} - 3)$
[$(4 \sqrt{3} - 2)(3 \sqrt{3} + 1)$
[$(2 \sqrt{5} + 1)(3 \sqrt{5} - 2)$

We hope you found this article helpful!