Simplify: 108 Y + 75 Y \sqrt{108y} + \sqrt{75y} 108 Y ​ + 75 Y ​ Show Your Work Here. Hint: To Add The Square Root Symbol ( □ \sqrt{\square} □ ​ ), Type root

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Understanding the Problem

To simplify the given expression, we need to combine the square roots of the two terms, $\sqrt{108y}$ and $\sqrt{75y}$. This involves finding a common factor that can be extracted from both terms, allowing us to rewrite the expression in a simpler form.

Breaking Down the Terms

Let's start by breaking down each term into its prime factors.

Prime Factorization of 108y

To find the prime factors of 108y, we can start by factoring 108:

108 = 2 × 2 × 3 × 3 × 3

Since 108y is a product of 108 and y, we can write it as:

108y = (2 × 2 × 3 × 3 × 3) × y

Prime Factorization of 75y

Next, let's factor 75:

75 = 3 × 5 × 5

Now, we can write 75y as:

75y = (3 × 5 × 5) × y

Simplifying the Expression

Now that we have the prime factorizations of both terms, we can rewrite the expression as:

$\sqrt{108y} + \sqrt{75y}$

= $\sqrt{(2 × 2 × 3 × 3 × 3) × y} + \sqrt{(3 × 5 × 5) × y}$

= $\sqrt{(2 × 2) × (3 × 3) × 3 × y} + \sqrt{(3) × (5 × 5) × y}$

= $\sqrt{(2^2) × (3^2) × 3 × y} + \sqrt{(3) × (5^2) × y}$

= $\sqrt{(2^2) × (3^2) × 3} × \sqrt{y} + \sqrt{(3) × (5^2)} × \sqrt{y}$

= $2 × 3 × \sqrt{3} × \sqrt{y} + 5 × \sqrt{3} × \sqrt{y}$

Combining Like Terms

Now that we have simplified each term, we can combine like terms by factoring out the common factor, $\sqrt{y}$:

= $(2 × 3 × \sqrt{3} + 5 × \sqrt{3}) × \sqrt{y}$

= $(6 × \sqrt{3} + 5 × \sqrt{3}) × \sqrt{y}$

= $(6 + 5) × \sqrt{3} × \sqrt{y}$

= $11 × \sqrt{3} × \sqrt{y}$

Final Simplified Expression

The final simplified expression is:

$\sqrt{108y} + \sqrt{75y} = 11\sqrt{3y}$

This expression cannot be simplified further, as there are no more common factors that can be extracted from both terms.

Conclusion

In this article, we simplified the given expression $\sqrt{108y} + \sqrt{75y}$ by breaking down each term into its prime factors and then combining like terms. The final simplified expression is $11\sqrt{3y}$, which cannot be simplified further.

Frequently Asked Questions

• Q: What is the prime factorization of 108? A: The prime factorization of 108 is 2 × 2 × 3 × 3 × 3.
• Q: What is the prime factorization of 75? A: The prime factorization of 75 is 3 × 5 × 5.
• Q: How do I simplify the expression $\sqrt{108y} + \sqrt{75y}$? A: To simplify the expression, break down each term into its prime factors and then combine like terms.

Additional Resources

• For more information on prime factorization, visit Wikipedia.
• For more information on simplifying expressions, visit Mathway.

Final Thoughts

Simplifying expressions is an essential skill in mathematics, and it requires a deep understanding of prime factorization and algebraic manipulation. By following the steps outlined in this article, you can simplify even the most complex expressions and arrive at a final answer.

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Frequently Asked Questions

Q: What is the prime factorization of 108?

A: The prime factorization of 108 is 2 × 2 × 3 × 3 × 3.

Q: What is the prime factorization of 75?

A: The prime factorization of 75 is 3 × 5 × 5.

Q: How do I simplify the expression $\sqrt{108y} + \sqrt{75y}$?

A: To simplify the expression, break down each term into its prime factors and then combine like terms.

Q: What is the final simplified expression for $\sqrt{108y} + \sqrt{75y}$?

A: The final simplified expression is $11\sqrt{3y}$.

Q: Can the expression $11\sqrt{3y}$ be simplified further?

A: No, the expression $11\sqrt{3y}$ cannot be simplified further, as there are no more common factors that can be extracted from both terms.

Q: What is the difference between simplifying an expression and solving an equation?

A: Simplifying an expression involves rewriting it in a simpler form, often by combining like terms or factoring out common factors. Solving an equation, on the other hand, involves finding the value of the variable that makes the equation true.

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

A: You should simplify an expression whenever possible, as it can make the expression easier to work with and understand. Simplifying expressions is an essential skill in mathematics, and it can help you solve problems more efficiently.

Q: Can I use a calculator to simplify expressions?

A: Yes, you can use a calculator to simplify expressions, but it's often more helpful to understand the underlying math and simplify expressions by hand. This can help you develop a deeper understanding of the math and make it easier to solve problems.

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

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

• Not combining like terms
• Not factoring out common factors
• Not checking for errors in the original expression
• Not using the correct order of operations

Q: How do I check my work when simplifying expressions?

A: To check your work when simplifying expressions, you should:

• Verify that you have combined like terms correctly
• Check that you have factored out common factors correctly
• Make sure that the final expression is in the simplest form possible
• Use a calculator to check your work, if necessary

Additional Resources

• For more information on prime factorization, visit Wikipedia.
• For more information on simplifying expressions, visit Mathway.
• For practice problems and exercises, visit Khan Academy.

Final Thoughts

Simplifying expressions is an essential skill in mathematics, and it requires a deep understanding of prime factorization and algebraic manipulation. By following the steps outlined in this article and practicing regularly, you can develop a strong foundation in simplifying expressions and become more confident in your math skills.