Simplify:$\sqrt{5}(\sqrt{2} + 4\sqrt{2}$\]

Feb 26, 2025 by ADMIN 43 views

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

Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently and accurately. One of the most common types of simplification is factoring out a common term from an expression. In this article, we will focus on simplifying the given expression $\sqrt{5}(\sqrt{2} + 4\sqrt{2})$ using the distributive property and other algebraic techniques.

Understanding the Expression

The given expression is a product of two terms: $\sqrt{5}$ and $(\sqrt{2} + 4\sqrt{2})$ . To simplify this expression, we need to apply the distributive property, which states that for any real numbers $a$ , $b$ , and $c$ , $a(b + c) = ab + ac$ . In this case, we can apply the distributive property to expand the product of $\sqrt{5}$ and $(\sqrt{2} + 4\sqrt{2})$ .

Applying the Distributive Property

Using the distributive property, we can expand the product as follows:

\sqrt{5}(\sqrt{2} + 4\sqrt{2}) = \sqrt{5}\sqrt{2} + \sqrt{5}4\sqrt{2}

Simplifying the Expression

Now that we have expanded the product, we can simplify the expression further by combining like terms. The expression can be rewritten as:

\sqrt{5}\sqrt{2} + 4\sqrt{5}\sqrt{2}

Factoring Out a Common Term

We can simplify the expression even further by factoring out a common term. In this case, the common term is $\sqrt{5}\sqrt{2}$ . We can factor this term out of both terms in the expression:

\sqrt{5}\sqrt{2} + 4\sqrt{5}\sqrt{2} = \sqrt{5}\sqrt{2}(1 + 4)

Simplifying the Factored Expression

Now that we have factored out the common term, we can simplify the expression further by evaluating the expression inside the parentheses:

\sqrt{5}\sqrt{2}(1 + 4) = \sqrt{5}\sqrt{2}(5)

Final Simplification

Finally, we can simplify the expression by combining the square roots:

\sqrt{5}\sqrt{2}(5) = \sqrt{5 \cdot 2 \cdot 5} = \sqrt{50}

Conclusion

In this article, we simplified the expression $\sqrt{5}(\sqrt{2} + 4\sqrt{2})$ using the distributive property and other algebraic techniques. We expanded the product, combined like terms, factored out a common term, and finally simplified the expression to $\sqrt{50}$ . This example demonstrates the importance of simplifying expressions in mathematics and how it can help us solve problems efficiently and accurately.

Additional Tips and Tricks

When simplifying expressions, always look for common terms that can be factored out.
Use the distributive property to expand products and combine like terms.
Simplify expressions by combining square roots and evaluating expressions inside parentheses.
Practice simplifying expressions regularly to develop your skills and build your confidence.

Common Mistakes to Avoid

Failing to factor out common terms can lead to unnecessary complexity and make it difficult to simplify expressions.
Not using the distributive property can result in incorrect expansions and make it challenging to combine like terms.
Not simplifying expressions can lead to errors and make it difficult to solve problems accurately.

Real-World Applications

Simplifying expressions is a crucial skill that has numerous real-world applications. In mathematics, simplifying expressions helps us solve problems efficiently and accurately. In science and engineering, simplifying expressions is essential for modeling complex systems and making predictions. In finance, simplifying expressions is critical for calculating interest rates and investment returns.

Final Thoughts

Simplifying expressions is a fundamental skill in mathematics that has numerous real-world applications. By understanding the distributive property, factoring out common terms, and combining like terms, we can simplify expressions efficiently and accurately. With practice and patience, anyone can develop the skills necessary to simplify expressions and solve problems confidently.

Introduction

In our previous article, we simplified the expression $\sqrt{5}(\sqrt{2} + 4\sqrt{2})$ using the distributive property and other algebraic techniques. In this article, we will answer some frequently asked questions (FAQs) related to simplifying expressions, including the one we discussed earlier.

Q&A

Q: What is the distributive property, and how is it used in simplifying expressions?

A: The distributive property is a fundamental concept in algebra that states that for any real numbers $a$ , $b$ , and $c$ , $a(b + c) = ab + ac$ . It is used to expand products and combine like terms, making it easier to simplify expressions.

Q: How do I simplify expressions with square roots?

A: To simplify expressions with square roots, you can combine the square roots by multiplying them together. For example, $\sqrt{5}\sqrt{2} = \sqrt{5 \cdot 2} = \sqrt{10}$ .

Q: What is the difference between factoring and simplifying expressions?

A: Factoring involves breaking down an expression into its constituent parts, while simplifying involves combining like terms and eliminating unnecessary complexity. Factoring is often used to solve equations, while simplifying is used to make expressions more manageable.

Q: Can I simplify expressions with variables?

A: Yes, you can simplify expressions with variables by combining like terms and eliminating unnecessary complexity. For example, $2x + 3x = 5x$ .

Q: How do I know when to simplify an expression?

A: You should simplify an expression when it becomes too complex or when you need to make it more manageable. Simplifying expressions can help you solve problems more efficiently and accurately.

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

A: Some common mistakes to avoid when simplifying expressions include:

Failing to factor out common terms
Not using the distributive property
Not simplifying expressions can lead to errors and make it difficult to solve problems accurately

Q: Can I simplify expressions with fractions?

A: Yes, you can simplify expressions with fractions by combining like terms and eliminating unnecessary complexity. For example, $\frac{1}{2} + \frac{1}{2} = \frac{2}{2} = 1$ .

Q: How do I simplify expressions with exponents?

A: To simplify expressions with exponents, you can combine like terms and eliminate unnecessary complexity. For example, $2^2 \cdot 2^3 = 2^{2+3} = 2^5$ .

Q: Can I simplify expressions with radicals?

A: Yes, you can simplify expressions with radicals by combining like terms and eliminating unnecessary complexity. For example, $\sqrt{5} + \sqrt{5} = 2\sqrt{5}$ .

Conclusion

Simplifying expressions is a crucial skill in mathematics that has numerous real-world applications. By understanding the distributive property, factoring out common terms, and combining like terms, we can simplify expressions efficiently and accurately. With practice and patience, anyone can develop the skills necessary to simplify expressions and solve problems confidently.

Additional Tips and Tricks

Always look for common terms that can be factored out.
Use the distributive property to expand products and combine like terms.
Simplify expressions by combining square roots and evaluating expressions inside parentheses.
Practice simplifying expressions regularly to develop your skills and build your confidence.

Common Mistakes to Avoid

Failing to factor out common terms can lead to unnecessary complexity and make it difficult to simplify expressions.
Not using the distributive property can result in incorrect expansions and make it challenging to combine like terms.
Not simplifying expressions can lead to errors and make it difficult to solve problems accurately.

Real-World Applications

Simplifying expressions is a fundamental skill that has numerous real-world applications. In mathematics, simplifying expressions helps us solve problems efficiently and accurately. In science and engineering, simplifying expressions is essential for modeling complex systems and making predictions. In finance, simplifying expressions is critical for calculating interest rates and investment returns.