Simplify The Expression: 9 125 9 \sqrt{125} 9 125 ​

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Introduction

Simplifying expressions involving square roots is a crucial skill in mathematics, particularly in algebra and geometry. In this article, we will focus on simplifying the expression $9 \sqrt{125}$, which involves a combination of multiplication and square root operations. We will break down the problem step by step, using various mathematical techniques to simplify the expression.

Understanding the Problem

The given expression is $9 \sqrt{125}$. To simplify this expression, we need to understand the properties of square roots and how they interact with multiplication. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16.

Breaking Down the Expression

To simplify the expression $9 \sqrt{125}$, we can start by breaking down the square root of 125. We can rewrite 125 as a product of its prime factors: $125 = 5^3$. This means that the square root of 125 can be expressed as the square root of $5^3$, which is equal to $5\sqrt{5}$.

Simplifying the Expression

Now that we have broken down the square root of 125, we can simplify the original expression. We can rewrite $9 \sqrt{125}$ as $9 \cdot 5\sqrt{5}$, using the property of multiplication that allows us to distribute the 9 to both the 5 and the square root of 5.

Applying the Distributive Property

The distributive property of multiplication states that for any numbers a, b, and c, the following equation holds: $a(b + c) = ab + ac$. We can apply this property to simplify the expression $9 \cdot 5\sqrt{5}$ by distributing the 9 to both the 5 and the square root of 5.

Simplifying Further

Using the distributive property, we can rewrite $9 \cdot 5\sqrt{5}$ as $45\sqrt{5}$. This is because the 9 multiplied by the 5 equals 45, and the 9 multiplied by the square root of 5 equals $9\sqrt{5}$.

Conclusion

In conclusion, we have simplified the expression $9 \sqrt{125}$ by breaking down the square root of 125 into its prime factors and applying the distributive property of multiplication. The simplified expression is $45\sqrt{5}$.

Final Answer

The final answer to the problem is $45\sqrt{5}$.

Additional Tips and Tricks

• When simplifying expressions involving square roots, it is often helpful to break down the square root into its prime factors.
• The distributive property of multiplication can be used to simplify expressions by distributing a number to both the numerator and the denominator.
• Practice simplifying expressions involving square roots to become more comfortable with the process.

Common Mistakes to Avoid

• Failing to break down the square root into its prime factors can lead to incorrect simplifications.
• Not applying the distributive property of multiplication can also result in incorrect simplifications.
• Not checking the final answer for accuracy can lead to errors.

Real-World Applications

Simplifying expressions involving square roots has many real-world, including:

• Calculating distances and heights in geometry and trigonometry
• Solving problems in physics and engineering
• Working with financial and economic data

Conclusion

In conclusion, simplifying expressions involving square roots is a crucial skill in mathematics, particularly in algebra and geometry. By breaking down the square root into its prime factors and applying the distributive property of multiplication, we can simplify expressions like $9 \sqrt{125}$ to their most basic form. With practice and patience, anyone can become proficient in simplifying expressions involving square roots.

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Introduction

In our previous article, we simplified the expression $9 \sqrt{125}$ by breaking down the square root of 125 into its prime factors and applying the distributive property of multiplication. In this article, we will answer some common questions related to simplifying expressions involving square roots.

Q&A

Q: What is the square root of 125?

A: The square root of 125 is $5\sqrt{5}$, because 125 can be rewritten as $5^3$.

Q: How do I simplify an expression involving a square root?

A: To simplify an expression involving a square root, you can break down the square root into its prime factors and apply the distributive property of multiplication.

Q: What is the distributive property of multiplication?

A: The distributive property of multiplication states that for any numbers a, b, and c, the following equation holds: $a(b + c) = ab + ac$. This means that you can distribute a number to both the numerator and the denominator of an expression.

Q: How do I apply the distributive property of multiplication to simplify an expression?

A: To apply the distributive property of multiplication, you can rewrite the expression as a product of two or more factors, and then distribute the number to each factor.

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

A: Some common mistakes to avoid include:

• Failing to break down the square root into its prime factors
• Not applying the distributive property of multiplication
• Not checking the final answer for accuracy

Q: What are some real-world applications of simplifying expressions involving square roots?

A: Simplifying expressions involving square roots has many real-world applications, including:

• Calculating distances and heights in geometry and trigonometry
• Solving problems in physics and engineering
• Working with financial and economic data

Q: How can I practice simplifying expressions involving square roots?

A: You can practice simplifying expressions involving square roots by working through examples and exercises, and by applying the techniques we discussed in this article to real-world problems.

Additional Tips and Tricks

• When simplifying expressions involving square roots, it is often helpful to break down the square root into its prime factors.
• The distributive property of multiplication can be used to simplify expressions by distributing a number to both the numerator and the denominator.
• Practice simplifying expressions involving square roots to become more comfortable with the process.

Common Misconceptions

• Some people may think that simplifying expressions involving square roots is only for advanced math students. However, simplifying expressions involving square roots is a fundamental skill that can be learned by anyone with practice and patience.
• Some people may think that simplifying expressions involving square roots is only for math problems. However, simplifying expressions involving square roots has many real-world applications, including physics, engineering, and finance.

Conclusion

In conclusion, simplifying expressions involving square roots is a crucial skill that can be learned by anyone with practice and patience. By breaking down the square root into its prime factors and applying the distributive property of multiplication, we can simplify expressions like $9 \sqrt{125}$ to their most basic form. With practice and patience, anyone can become proficient in simplifying expressions involving square roots.

Final Answer

The final answer to the problem is $45\sqrt{5}$.

Additional Resources

• For more information on simplifying expressions involving square roots, check out our previous article on the topic.
• For practice exercises and examples, try working through the problems in a math textbook or online resource.
• For real-world applications of simplifying expressions involving square roots, try searching online for examples and case studies.