Simplify Your Math Troubles: The Easy Way to Solve Square Root of 3 Times Square Root of 21!

Simplify square root of 3 times square root of 21 with ease using our step-by-step guide. Get rid of the complexity and solve it in seconds!

If you have ever come across the expression simplify square root of 3 times square root of 21, then you know how challenging it can be to solve. This mathematical expression may seem daunting, but with a little bit of guidance, it can become a lot easier to handle. In this article, we will delve deeper into the world of square roots and explore different ways to simplify square root of 3 times square root of 21.

Before we begin, let us first understand what square roots are. A square root is a number that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because 4 x 4 = 16. Similarly, the square root of 9 is 3 because 3 x 3 = 9. Square roots are denoted by the symbol √.

Now, let us move on to the expression we are trying to simplify - square root of 3 times square root of 21. One way to simplify this expression is by using the product rule of square roots. According to this rule, square root of a x square root of b is equal to the square root of a x b. Using this rule, we can simplify the expression to square root of 63.

Although we have simplified the expression, it is still not in its simplest form. We can further simplify square root of 63 by breaking it down into its prime factors. Prime factors are numbers that can only be divided by 1 and themselves. To find the prime factors of 63, we can divide it by the smallest prime number, which is 2. We get 31.5, which is not a whole number. Therefore, we divide 63 by the next smallest prime number, which is 3. This gives us 21. After dividing 21 by 3, we get 7. We cannot divide 7 any further, so the prime factors of 63 are 3, 3, and 7.

Now that we know the prime factors of 63, we can rewrite square root of 63 as square root of 3 x 3 x 7. Using the product rule of square roots again, we can simplify this expression to 3√7. Therefore, the simplified form of square root of 3 times square root of 21 is 3√7.

It is important to note that there are other methods to simplify square root of 3 times square root of 21. For example, we can use the distributive property of multiplication to rewrite the expression as square root of 3 x 21. We can then simplify this expression using the same method as above. However, the product rule of square roots is often the easiest and most efficient way to simplify expressions of this kind.

In conclusion, simplifying square root of 3 times square root of 21 may seem like a daunting task, but with the right tools and techniques, it can become much easier to handle. By using the product rule of square roots and breaking down the expression into its prime factors, we can simplify square root of 3 times square root of 21 to 3√7. Understanding how to simplify expressions like these is an essential skill for anyone studying mathematics or related fields.

Introduction

Mathematics is an essential part of our life, and the knowledge of basic mathematical concepts is crucial for every individual. When it comes to algebraic expressions, simplifying them can be a challenging task. In this article, we will discuss how to simplify the square root of 3 times the square root of 21.

What is Square Root?

Square root is a mathematical operation that determines the value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 times 4 equals 16. The symbol for square root is √.

Understanding the Problem

The problem we are discussing is the simplification of the square root of 3 times the square root of 21. To solve this problem, we need to use the properties of square roots.

Properties of Square Roots

Product Property

The product property of square roots states that the square root of a product is equal to the product of the square roots of the factors. For example, the square root of 25 times the square root of 4 is equal to the square root of 100.

Simplification Property

The simplification property of square roots states that the square root of a perfect square is a whole number. For example, the square root of 25 is 5, which is a whole number. However, the square root of 21 is not a perfect square, so we need to simplify it further.

Simplification of Square Root of 21

To simplify the square root of 21, we can break it down into its prime factors. The prime factorization of 21 is 3 x 7. Therefore, the square root of 21 can be simplified as the square root of 3 times the square root of 7.

Applying the Product Property

Now that we have simplified the square root of 21, we can apply the product property of square roots to simplify the expression. The product property states that the square root of a product is equal to the product of the square roots of the factors. Therefore, the square root of 3 times the square root of 21 is equal to the square root of 3 times the square root of 3 times the square root of 7.

Final Simplification

Now, we can simplify the expression further by using the simplification property of square roots. The simplification property states that the square root of a perfect square is a whole number. Since the square root of 3 times the square root of 3 is equal to the square root of 9, which is a perfect square, we can simplify it as 3. Therefore, the final simplification of the expression is 3 times the square root of 7.

Conclusion

In conclusion, simplifying algebraic expressions involving square roots requires an understanding of the properties of square roots. By using the product property and simplification property of square roots, we can simplify the square root of 3 times the square root of 21 as 3 times the square root of 7.

Understanding the Basics of Square Roots

As we begin to explore the concept of simplifying square roots of 3 times square roots of 21, it's essential to have an understanding of the basics of square roots. Square roots are used to determine the value of a number when it's multiplied by itself. For instance, the square root of 4 is 2, as 2 multiplied by 2 equals 4. Similarly, the square root of 9 is 3, and the square root of 16 is 4.

Expressing the Square Root of 3

The square root of 3 is often expressed as √3. However, this can pose a challenge when it comes to simplifying expressions that involve this value. Therefore, we need to follow a systematic method to simplify such expressions.

The Importance of Factoring

One way to simplify expressions involving square roots is to factor the numbers into their prime factors. This can help us identify numbers that can be simplified or eliminated. Factoring means breaking down a number into its smaller factors. For example, to factor 12, we can break it down to 2 x 2 x 3.

Factors of 21

To simplify square roots of 3 times square roots of 21, we first need to identify the factors of 21. These are 1, 3, 7, and 21. By identifying these factors, we can proceed with simplifying the expression.

Multiplying Square Roots

When multiplying square roots, we can simplify the expression by multiplying the values inside the roots. In this case, we would multiply √3 by √21. It is important to note that the product of two square roots is the same as the square root of their product.

Simplifying the Expression

To simplify the expression of square roots of 3 times square roots of 21, we need to break down the value inside the square root of 21. This can be done by using the prime factorization method. The prime factorization of 21 is 3 x 7. These numbers can be pulled out of the square root and combined with the square root of 3.

Simplifying Further

Once we have factored the value inside the square root, we can simplify it further by multiplying the values outside of the roots. In this case, we have √3 x √(3x7). By multiplying these values, we get √63.

The Final Result

After simplifying as much as possible, the expression of square roots of 3 times square roots of 21 can be written as √63 or 3√7. Both expressions are equivalent and simplified.

Checking Our Work

It's always important to double-check our work, especially when dealing with complex expressions. To ensure our simplified expression is correct, we can use a calculator to determine the value of both expressions and compare them. This is an important step in ensuring that we have simplified the expression correctly.

The Story of Simplifying Square Root of 3 Times Square Root of 21

The Confusion

As a student, I always found myself struggling with math. And one day, my teacher introduced us to the concept of simplifying square roots. It was all going great until we reached the problem of Simplify Square Root of 3 Times Square Root of 21.

The Empathic Voice

I could feel the frustration building up inside me as I tried to wrap my head around the concept. But then, my teacher noticed my confusion and decided to help me out. She spoke to me in an empathic voice, assuring me that it was okay to struggle and that she would guide me through the process.

The Explanation

First, she explained that the square root of 21 can be broken down into the product of two numbers - 3 and 7. Therefore, Simplify Square Root of 3 Times Square Root of 21 is the same as Simplify Square Root of 3 Times Square Root of 3 Times Square Root of 7.

Next, she showed me how to simplify the square roots of 3 by taking out the factors of the perfect squares. Thus, Simplify Square Root of 3 Times Square Root of 3 Times Square Root of 7 becomes 3 Times Square Root of 7.

The Clarity

With her guidance, I finally understood the concept of simplifying square roots. I felt a wave of clarity wash over me as I solved the problem with ease. My teacher's empathic voice and tone had made all the difference in my understanding of the topic.

Table Information

Here is a table showing the breakdown of Simplify Square Root of 3 Times Square Root of 21:

Problem	Solution
Simplify Square Root of 21	Square Root of 3 Times Square Root of 7
Simplify Square Root of 3 Times Square Root of 3 Times Square Root of 7	3 Times Square Root of 7

As shown in the table, Simplify Square Root of 3 Times Square Root of 21 can be simplified to 3 Times Square Root of 7.

Simplify Square Root Of 3 Times Square Root Of 21

Dear blog visitors,

Thank you for taking the time to read this article about simplifying square root of 3 times square root of 21. We hope that this has been a helpful and informative read for you, and that you now have a better understanding of how to simplify expressions involving square roots.

When it comes to simplifying square roots, there are a few key rules that you need to keep in mind. First and foremost, you need to remember that you can only combine square roots if they have the same radicand (the number under the radical sign).

In the case of square root of 3 times square root of 21, we cannot simply multiply the two values together to get a single simplified expression. Instead, we need to use one of the rules of exponents, which tells us that the product of two square roots is equal to the square root of their product.

So, to simplify square root of 3 times square root of 21, we need to first find the product of these two values:

√3 x √21 = √(3 x 21) = √63

Now, we need to simplify the square root of 63. To do this, we need to factor 63 into its prime factors:

63 = 3 x 3 x 7

Now, we can rewrite the square root of 63 as the square root of 3 x 3 x 7:

√63 = √(3 x 3 x 7) = √3 x √3 x √7

Finally, we can simplify this expression by combining the two square roots of 3:

√3 x √3 x √7 = 3√7

So, the simplified expression for square root of 3 times square root of 21 is 3√7.

We hope that this step-by-step guide has been helpful for you in understanding how to simplify expressions involving square roots. If you have any further questions or would like more information on this topic, please feel free to leave a comment below.

Thank you again for reading, and we wish you all the best in your mathematical endeavors!

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