Simplify The Expression: 18 60 2 3 \frac{18 \sqrt{60}}{2 \sqrt{3}} 2 3 ​ 18 60 ​ ​

$$

Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the techniques involved in simplifying expressions that contain square roots. In this article, we will focus on simplifying the expression $\frac{18 \sqrt{60}}{2 \sqrt{3}}$. We will break down the expression into smaller parts, simplify each part, and then combine them to obtain the final simplified expression.

Understanding the Expression

The given expression is $\frac{18 \sqrt{60}}{2 \sqrt{3}}$. To simplify this expression, we need to understand the properties of square roots and how to manipulate them. 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.

Simplifying the Square Roots

The expression contains two square roots: $\sqrt{60}$ and $\sqrt{3}$. To simplify these square roots, we need to find the largest perfect square that divides each number. A perfect square is a number that can be expressed as the product of an integer with itself. For example, 16 is a perfect square because it can be expressed as 4 multiplied by 4.

Simplifying $\sqrt{60}$

To simplify $\sqrt{60}$, we need to find the largest perfect square that divides 60. We can start by finding the prime factorization of 60:

60 = 2 × 2 × 3 × 5

We can see that 60 can be expressed as 2 × 2 × 3 × 5. The largest perfect square that divides 60 is 2 × 2, which equals 4. Therefore, we can simplify $\sqrt{60}$ as follows:

$\sqrt{60}$ = $\sqrt{2 × 2 × 3 × 5}$

= $\sqrt{2^2 × 3 × 5}$

= 2$\sqrt{3 × 5}$

= 2$\sqrt{15}$

Simplifying $\sqrt{3}$

To simplify $\sqrt{3}$, we can see that it is already a perfect square, because 3 can be expressed as 1 multiplied by 3. Therefore, we can simplify $\sqrt{3}$ as follows:

$\sqrt{3}$ = $\sqrt{1 × 3}$

= $\sqrt{3}$

Simplifying the Expression

Now that we have simplified the square roots, we can simplify the expression $\frac{18 \sqrt{60}}{2 \sqrt{3}}$. We can start by substituting the simplified square roots into the expression:

$\frac{18 \sqrt{60}}{2 \sqrt{3}}$

= $\frac{18 \sqrt{2^2 × 3 × 5}}{2 \sqrt{3}}$

= $\frac{18 × 2 \sqrt{3 × 5}}{2 \sqrt{3}}$

= $\frac{36 \sqrt{3 × 5}}{2 \sqrt{3}}$

= $\frac{36 \sqrt{15}}{2 \sqrt{3}}$

Canceling Out Common Factors

Now that we have simplified the expression, we can cancel out common factors. We can see that both the numerator and the denominator contain a factor of $\sqrt{3}$. We can cancel out this factor as follows:

$\frac{36 \sqrt{15}}{2 \sqrt{3}}$

= $\frac{36 \sqrt{15}}{2 \sqrt{3}} × \frac{\sqrt{3}}{\sqrt{3}}$

= $\frac{36 \sqrt{15} \sqrt{3}}{2 \sqrt{3} \sqrt{3}}$

= $\frac{36 \sqrt{45}}{6}$

= $\frac{36 \sqrt{9 × 5}}{6}$

= $\frac{36 \sqrt{9} \sqrt{5}}{6}$

= $\frac{36 × 3 \sqrt{5}}{6}$

= $\frac{108 \sqrt{5}}{6}$

= $18 \sqrt{5}$

Conclusion

In this article, we simplified the expression $\frac{18 \sqrt{60}}{2 \sqrt{3}}$ by breaking it down into smaller parts, simplifying each part, and then combining them to obtain the final simplified expression. We used the properties of square roots and manipulated them to simplify the expression. The final simplified expression is $18 \sqrt{5}$.

Final Answer

The final answer is: $\boxed{18 \sqrt{5}}$

$$

Introduction

In our previous article, we simplified the expression $\frac{18 \sqrt{60}}{2 \sqrt{3}}$ to $18 \sqrt{5}$. However, we understand that simplifying algebraic expressions can be a challenging task, and many readers may have questions about the process. In this article, we will address some of the most frequently asked questions about simplifying the expression $\frac{18 \sqrt{60}}{2 \sqrt{3}}$.

Q&A

Q: What is the first step in simplifying the expression $\frac{18 \sqrt{60}}{2 \sqrt{3}}$?

A: The first step in simplifying the expression $\frac{18 \sqrt{60}}{2 \sqrt{3}}$ is to simplify the square roots. We need to find the largest perfect square that divides each number and then simplify the square roots.

Q: How do I simplify $\sqrt{60}$?

A: To simplify $\sqrt{60}$, we need to find the largest perfect square that divides 60. We can start by finding the prime factorization of 60:

60 = 2 × 2 × 3 × 5

We can see that 60 can be expressed as 2 × 2 × 3 × 5. The largest perfect square that divides 60 is 2 × 2, which equals 4. Therefore, we can simplify $\sqrt{60}$ as follows:

$\sqrt{60}$ = $\sqrt{2 × 2 × 3 × 5}$

= $\sqrt{2^2 × 3 × 5}$

= 2$\sqrt{3 × 5}$

= 2$\sqrt{15}$

Q: How do I simplify $\sqrt{3}$?

A: To simplify $\sqrt{3}$, we can see that it is already a perfect square, because 3 can be expressed as 1 multiplied by 3. Therefore, we can simplify $\sqrt{3}$ as follows:

$\sqrt{3}$ = $\sqrt{1 × 3}$

= $\sqrt{3}$

Q: What is the next step in simplifying the expression $\frac{18 \sqrt{60}}{2 \sqrt{3}}$?

A: The next step in simplifying the expression $\frac{18 \sqrt{60}}{2 \sqrt{3}}$ is to substitute the simplified square roots into the expression and then cancel out common factors.

Q: How do I cancel out common factors in the expression $\frac{36 \sqrt{15}}{2 \sqrt{3}}$?

A: To cancel out common factors in the expression $\frac{36 \sqrt{15}}{2 \sqrt{3}}$, we can see that both the numerator and the denominator contain a factor of $\sqrt{3}$. We can cancel out this factor as follows:

$\frac{36 \sqrt{15}}{2 \sqrt{3}}$

= $\frac{36 \sqrt{15}}{2 \sqrt{3}} × \frac{\sqrt{3}}{\sqrt{3}}$

= $\frac{36 \sqrt{15} \sqrt{3}}{2 \sqrt{3} \sqrt{3}}$

= $\frac{36 \sqrt{45}}{6}$

= $\frac{36 \sqrt{9 × 5}}{6}$

= $\frac{36 \sqrt{9} \sqrt{5}}{6}$

= $\frac{36 × 3 \sqrt{5}}{6}$

= $\frac{108 \sqrt{5}}{6}$

= $18 \sqrt{5}$

Q: What is the final simplified expression?

A: The final simplified expression is $18 \sqrt{5}$.

Conclusion

In this article, we addressed some of the most frequently asked questions about simplifying the expression $\frac{18 \sqrt{60}}{2 \sqrt{3}}$. We provided step-by-step instructions on how to simplify the square roots, substitute the simplified square roots into the expression, and cancel out common factors. The final simplified expression is $18 \sqrt{5}$.

Final Answer

The final answer is: $\boxed{18 \sqrt{5}}$