Simplify The Expression:$\[ \frac{4 \sqrt{9}}{5 \sqrt{6}} \\]

Introduction

In mathematics, simplifying expressions is an essential skill that helps us solve problems efficiently and accurately. In this article, we will focus on simplifying the given expression: $\frac{4 \sqrt{9}}{5 \sqrt{6}}$. We will break down the steps involved in simplifying this expression and provide a clear understanding of the concepts used.

Understanding the Expression

The given expression is a fraction with square roots in the numerator and denominator. To simplify this expression, we need to understand the properties of square roots and how to manipulate them.

Properties of Square Roots

A 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. We can represent square roots using the symbol $\sqrt{}$. For example, $\sqrt{16}$ represents the square root of 16.

Simplifying the Numerator

The numerator of the given expression is $4 \sqrt{9}$. To simplify this, we need to find the square root of 9. Since $\sqrt{9}$ equals 3, we can rewrite the numerator as $4 \times 3 = 12$.

Simplifying the Denominator

The denominator of the given expression is $5 \sqrt{6}$. To simplify this, we need to find the square root of 6. Since $\sqrt{6}$ is an irrational number, we cannot simplify it further. However, we can rewrite the denominator as $5 \times \sqrt{6}$.

Simplifying the Expression

Now that we have simplified the numerator and denominator, we can rewrite the expression as $\frac{12}{5 \times \sqrt{6}}$. To simplify this further, we can multiply the numerator and denominator by the square root of 6. This gives us:

$\frac{12 \times \sqrt{6}}{5 \times \sqrt{6} \times \sqrt{6}}$

Rationalizing the Denominator

The denominator of the expression contains two square roots: $\sqrt{6}$ and $\sqrt{6}$. We can simplify this by multiplying the numerator and denominator by the square root of 6. This gives us:

$\frac{12 \times \sqrt{6} \times \sqrt{6}}{5 \times \sqrt{6} \times \sqrt{6} \times \sqrt{6}}$

Simplifying the Expression Further

Now that we have rationalized the denominator, we can simplify the expression further. We can rewrite the numerator as $12 \times 6 = 72$, and the denominator as $5 \times 6 \times \sqrt{6} = 30 \times \sqrt{6}$. This gives us:

$\frac{72}{30 \times \sqrt{6}}$

Final Simplification

To simplify the expression further, we can divide the numerator and denominator by their greatest common divisor, which is 6. This gives us:

$\frac{12}{5 \times \sqrt{6}}$

Conclusion

In this article, we simplified the given expression $\frac{4 \sqrt{9}}{5 \sqrt{6}}$ using the properties of square roots and rationalizing the denominator. We broke down the steps involved in simplifying the expression and provided a clear understanding of the concepts used. By following these steps, you can simplify similar expressions and become more confident in your mathematical skills.

Common Mistakes to Avoid

When simplifying expressions with square roots, it's essential to avoid common mistakes. Here are a few:

• Not simplifying the numerator and denominator separately: Make sure to simplify the numerator and denominator separately before combining them.
• Not rationalizing the denominator: Rationalizing the denominator is crucial when simplifying expressions with square roots.
• Not checking for common factors: Always check for common factors between the numerator and denominator before simplifying the expression.

Practice Problems

To practice simplifying expressions with square roots, try the following problems:

• $\frac{3 \sqrt{16}}{4 \sqrt{9}}$
• $\frac{2 \sqrt{25}}{5 \sqrt{4}}$
• $\frac{5 \sqrt{36}}{3 \sqrt{49}}$

Conclusion

$$

Introduction

In our previous article, we simplified the expression $\frac{4 \sqrt{9}}{5 \sqrt{6}}$ using the properties of square roots and rationalizing the denominator. In this article, we will provide a Q&A guide to help you understand the concepts and steps involved in simplifying expressions with square roots.

Q: What is the difference between a square root and a rational number?

A: A square root 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. A rational number is a number that can be expressed as the ratio of two integers. For example, 3/4 is a rational number.

Q: How do I simplify an expression with a square root in the numerator?

A: To simplify an expression with a square root in the numerator, you need to find the square root of the number inside the square root. For example, if the expression is $\frac{4 \sqrt{9}}{5 \sqrt{6}}$, you can simplify the numerator by finding the square root of 9, which is 3.

Q: How do I rationalize the denominator of an expression?

A: To rationalize the denominator of an expression, you need to multiply the numerator and denominator by the square root of the number inside the square root. For example, if the expression is $\frac{4 \sqrt{9}}{5 \sqrt{6}}$, you can rationalize the denominator by multiplying the numerator and denominator by $\sqrt{6}$.

Q: What is the greatest common divisor (GCD) and how do I use it to simplify an expression?

A: The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. To simplify an expression, you can divide the numerator and denominator by their GCD. For example, if the expression is $\frac{12}{30}$, you can simplify it by dividing both numbers by 6, which is their GCD.

Q: How do I simplify an expression with multiple square roots in the denominator?

A: To simplify an expression with multiple square roots in the denominator, you need to rationalize the denominator by multiplying the numerator and denominator by the product of the square roots. For example, if the expression is $\frac{4 \sqrt{9}}{5 \sqrt{6} \times \sqrt{6}}$, you can rationalize the denominator by multiplying the numerator and denominator by $\sqrt{6}$.

Q: Can I simplify an expression with a square root in the denominator?

A: Yes, you can simplify an expression with a square root in the denominator by rationalizing the denominator. For example, if the expression is $\frac{4}{\sqrt{6}}$, you can rationalize the denominator by multiplying the numerator and denominator by $\sqrt{6}$.

Q: How do I check if an expression is simplified?

A: To check if an expression is simplified, you need to make sure that there are no common factors between the numerator and denominator. You can also check if the expression can be simplified further by dividing both numbers by their GCD.

Conclusion

Simplifying expressions with square roots requires a clear understanding of the properties of square roots and how to manipulate them. By following the steps outlined in this article and practicing with Q&A, you can become more confident in your mathematical skills and simplify expressions like $\frac{4 \sqrt{9}}{5 \sqrt{6}}$.

Practice Problems

To practice simplifying expressions with square roots, try the following problems:

• $\frac{3 \sqrt{16}}{4 \sqrt{9}}$
• $\frac{2 \sqrt{25}}{5 \sqrt{4}}$
• $\frac{5 \sqrt{36}}{3 \sqrt{49}}$

Common Mistakes to Avoid

When simplifying expressions with square roots, it's essential to avoid common mistakes. Here are a few:

• Not simplifying the numerator and denominator separately: Make sure to simplify the numerator and denominator separately before combining them.
• Not rationalizing the denominator: Rationalizing the denominator is crucial when simplifying expressions with square roots.
• Not checking for common factors: Always check for common factors between the numerator and denominator before simplifying the expression.