Simplify The Expression: $\sqrt{\frac{10}{49}}$

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Introduction

Simplifying expressions involving square roots is a fundamental concept in mathematics, particularly in algebra and geometry. In this article, we will focus on simplifying the expression $\sqrt{\frac{10}{49}}$. This involves understanding the properties of square roots, simplifying fractions, and applying mathematical operations to arrive at the final result.

Understanding 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. In mathematical notation, this is represented as $\sqrt{16} = 4$. The square root of a number can be either positive or negative, and both values are considered square roots of the number.

Simplifying Fractions

To simplify the expression $\sqrt{\frac{10}{49}}$, we need to start by simplifying the fraction inside the square root. The fraction $\frac{10}{49}$ can be simplified by finding the greatest common divisor (GCD) of the numerator and denominator. The GCD of 10 and 49 is 1, which means that the fraction cannot be simplified further.

Applying Mathematical Operations

Now that we have simplified the fraction, we can apply mathematical operations to simplify the expression. We can start by breaking down the fraction into its prime factors. The prime factorization of 10 is $2 \times 5$, and the prime factorization of 49 is $7^2$. We can rewrite the fraction as $\frac{2 \times 5}{7^2}$.

Simplifying the Expression

Now that we have broken down the fraction into its prime factors, we can simplify the expression by taking the square root of the numerator and denominator separately. The square root of $2 \times 5$ is $\sqrt{2} \times \sqrt{5}$, and the square root of $7^2$ is $7$. We can rewrite the expression as $\frac{\sqrt{2} \times \sqrt{5}}{7}$.

Final Result

The final result of simplifying the expression $\sqrt{\frac{10}{49}}$ is $\frac{\sqrt{2} \times \sqrt{5}}{7}$. This can be further simplified by combining the square roots of 2 and 5 into a single square root. The square root of 10 is $\sqrt{2} \times \sqrt{5}$, so we can rewrite the expression as $\frac{\sqrt{10}}{7}$.

Conclusion

Simplifying expressions involving square roots requires a thorough understanding of mathematical operations and properties. By breaking down the fraction into its prime factors and applying mathematical operations, we can simplify the expression $\sqrt{\frac{10}{49}}$ to $\frac{\sqrt{10}}{7}$. This result demonstrates the importance of simplifying expressions in mathematics and provides a foundation for further mathematical exploration.

Additional Examples

• Simplify the expression $\sqrt{\frac{25}{36}}$.
• Simplify the expression $\sqrt{\frac{64}{81}}$.
• Simplify the expression $\sqrt{\frac{9}{16}}$.

Step-by-Step Solution

To simplify the expression $\sqrt{\frac{10}{49}}$, follow these steps:

1. Simplify the fraction inside the square root by finding the greatest common divisor (GCD) of the numerator and denominator.
2. Break down the fraction into its prime factors.
3. Take the square root of the numerator and denominator separately.
4. Combine the square roots of the numerator and denominator into a single square root.
5. Simplify the resulting expression.

Frequently Asked Questions

• What is the square root of a number?
• How do you simplify a fraction?
• What is the greatest common divisor (GCD) of two numbers?
• How do you break down a fraction into its prime factors?
• How do you simplify an expression involving square roots?

Glossary of Terms

• Square root: A value that, when multiplied by itself, gives the original number.
• Greatest common divisor (GCD): The largest number that divides two or more numbers without leaving a remainder.
• Prime factorization: The process of breaking down a number into its prime factors.
• Simplifying an expression: Reducing an expression to its simplest form by applying mathematical operations.
  

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Introduction

In our previous article, we explored the concept of simplifying expressions involving square roots, focusing on the expression $\sqrt{\frac{10}{49}}$. We broke down the fraction into its prime factors, applied mathematical operations, and arrived at the final result of $\frac{\sqrt{10}}{7}$. In this article, we will delve into a Q&A format, addressing common questions and providing additional insights into simplifying expressions involving square roots.

Q&A

Q: What is the square root of a number?

A: 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.

Q: How do you simplify a fraction?

A: To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and denominator. The GCD is the largest number that divides both numbers without leaving a remainder. Once you have found the GCD, you can divide both the numerator and denominator by the GCD to simplify the fraction.

Q: What is the greatest common divisor (GCD) of two numbers?

A: The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder. For example, the GCD of 10 and 49 is 1, because 1 is the largest number that divides both 10 and 49 without leaving a remainder.

Q: How do you break down a fraction into its prime factors?

A: To break down a fraction into its prime factors, you need to find the prime factors of the numerator and denominator. The prime factors of a number are the prime numbers that multiply together to give the original number. For example, the prime factorization of 10 is $2 \times 5$, and the prime factorization of 49 is $7^2$.

Q: How do you simplify an expression involving square roots?

A: To simplify an expression involving square roots, you need to follow these steps:

1. Simplify the fraction inside the square root by finding the greatest common divisor (GCD) of the numerator and denominator.
2. Break down the fraction into its prime factors.
3. Take the square root of the numerator and denominator separately.
4. Combine the square roots of the numerator and denominator into a single square root.
5. Simplify the resulting expression.

Q: What is the final result of simplifying the expression $\sqrt{\frac{10}{49}}$?

A: The final result of simplifying the expression $\sqrt{\frac{10}{49}}$ is $\frac{\sqrt{10}}{7}$.

Q: Can you provide additional examples of simplifying expressions involving square roots?

A: Yes, here are a few examples:

• Simplify the expression $\sqrt{\frac{25}{36}}$.
• Simplify the expression $\sqrt{\frac{64}{81}}$.
• Simplify the expression $\sqrt{\frac{9}{16}}$.

Conclusion

Simplifying expressions involving square roots requires a thorough understanding of mathematical operations and properties. By breaking down the fraction into its prime factors and applying mathematical operations, we can simplify the expression $\sqrt{\frac{10}{49}}$ to $\frac{\sqrt{10}}{7}$. This result demonstrates the importance of simplifying expressions in mathematics and provides a foundation for further mathematical exploration.

Additional Resources

• Simplifying Expressions Involving Square Roots
• Mathematical Operations and Properties
• Prime Factorization

Frequently Asked Questions

• What is the square root of a number?
• How do you simplify a fraction?
• What is the greatest common divisor (GCD) of two numbers?
• How do you break down a fraction into its prime factors?
• How do you simplify an expression involving square roots?

Glossary of Terms

• Square root: A value that, when multiplied by itself, gives the original number.
• Greatest common divisor (GCD): The largest number that divides two or more numbers without leaving a remainder.
• Prime factorization: The process of breaking down a number into its prime factors.
• Simplifying an expression: Reducing an expression to its simplest form by applying mathematical operations.