Simplify.9) 48 \sqrt{48} 48 ​

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Introduction

Simplifying square roots is an essential skill in mathematics, particularly in algebra and geometry. It involves expressing a square root in its simplest form, which can be achieved by factoring the number inside the square root sign into its prime factors. In this article, we will simplify the square root of 48, which is denoted as $\sqrt{48}$. We will break down the process step by step and provide a clear explanation of each step.

Step 1: Factorize the Number Inside the Square Root Sign

To simplify $\sqrt{48}$, we need to factorize the number 48 into its prime factors. The prime factorization of 48 is $2^4 \times 3$. This means that 48 can be expressed as the product of 2 raised to the power of 4 and 3.

Step 2: Identify the Perfect Square Factor

Next, we need to identify the perfect square factor in the prime factorization of 48. A perfect square factor is a factor that can be expressed as the square of an integer. In this case, the perfect square factor is $2^4$, which is equal to 16.

Step 3: Simplify the Square Root

Now that we have identified the perfect square factor, we can simplify the square root of 48. We can rewrite $\sqrt{48}$ as $\sqrt{2^4 \times 3}$, which is equal to $\sqrt{16 \times 3}$. Since 16 is a perfect square, we can take the square root of 16, which is equal to 4. Therefore, $\sqrt{48}$ can be simplified as $4\sqrt{3}$.

Example

Let's consider an example to illustrate the concept of simplifying square roots. Suppose we want to simplify $\sqrt{72}$. We can factorize 72 into its prime factors, which is $2^3 \times 3^2$. The perfect square factor in this case is $3^2$, which is equal to 9. Therefore, $\sqrt{72}$ can be simplified as $3\sqrt{8}$.

Real-World Applications

Simplifying square roots has numerous real-world applications in various fields, including physics, engineering, and computer science. For instance, in physics, the square root of a quantity is often used to represent the magnitude of a vector. In engineering, the square root of a quantity is used to represent the magnitude of a force or a torque. In computer science, the square root of a quantity is used to represent the magnitude of a signal or a noise.

Conclusion

In conclusion, simplifying square roots is an essential skill in mathematics that has numerous real-world applications. By breaking down the process step by step and identifying the perfect square factor, we can simplify the square root of a number. In this article, we simplified the square root of 48, which is denoted as $\sqrt{48}$. We also provided an example to illustrate the concept of simplifying square roots and discussed the real-world applications of this concept.

Frequently Asked Questions

• Q: What is the square root of 48? A: The square root of 48 is $4\sqrt{3}$.
• Q: How do I simplify the square root of a number? A: To simplify the square root of a number, you need to factorize the number into its prime factors and identify the perfect square factor.
• Q: What is the perfect square factor? A: The perfect square factor is a factor that can be expressed as the square of an integer.

Further Reading

If you want to learn more about simplifying square roots, I recommend checking out the following resources:

• Khan Academy: Simplifying Square Roots
• Mathway: Simplifying Square Roots
• Wolfram Alpha: Simplifying Square Roots

References

• "Algebra" by Michael Artin
• "Geometry" by Michael Spivak
• "Calculus" by Michael Spivak
  

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Introduction

In our previous article, we simplified the square root of 48, which is denoted as $\sqrt{48}$. We broke down the process step by step and provided a clear explanation of each step. In this article, we will answer some frequently asked questions related to simplifying square roots.

Q&A

Q: What is the square root of 48?

A: The square root of 48 is $4\sqrt{3}$.

Q: How do I simplify the square root of a number?

A: To simplify the square root of a number, you need to factorize the number into its prime factors and identify the perfect square factor.

Q: What is the perfect square factor?

A: The perfect square factor is a factor that can be expressed as the square of an integer.

Q: How do I identify the perfect square factor?

A: To identify the perfect square factor, you need to look for a factor that can be expressed as the square of an integer. For example, if you have a number like 16, you can express it as $2^4$, which is a perfect square factor.

Q: Can I simplify the square root of a number that has a perfect square factor?

A: Yes, you can simplify the square root of a number that has a perfect square factor. For example, if you have a number like 48, you can express it as $2^4 \times 3$, which has a perfect square factor of $2^4$. Therefore, the square root of 48 can be simplified as $4\sqrt{3}$.

Q: Can I simplify the square root of a number that has no perfect square factor?

A: No, you cannot simplify the square root of a number that has no perfect square factor. For example, if you have a number like 50, you cannot simplify its square root because it has no perfect square factor.

Q: How do I simplify the square root of a number that has multiple perfect square factors?

A: To simplify the square root of a number that has multiple perfect square factors, you need to identify all the perfect square factors and simplify the square root accordingly. For example, if you have a number like 48, you can express it as $2^4 \times 3$, which has two perfect square factors: $2^4$ and $3^0$. Therefore, the square root of 48 can be simplified as $4\sqrt{3}$.

Q: Can I simplify the square root of a negative number?

A: No, you cannot simplify the square root of a negative number. The square root of a negative number is an imaginary number, which cannot be simplified.

Q: Can I simplify the square root of a fraction?

A: Yes, you can simplify the square root of a fraction. For example, if you have a fraction like $\frac{1}{4}$, you can simplify its square root as $\frac{1}{2}$.

Example Problems

Problem 1

Simplify the square root of 72.

Solution:

To simplify the square root of 72, we need to factorize 72 into its prime factors, which is $2^3 \times 3^2$. The perfect square factor in this case is $3^2$, which is equal to 9. Therefore, the square root of 72 can be simplified as $3\sqrt{8}$.

Problem 2

Simplify the square root of 50.

Solution:

To simplify the square root of 50, we need to factorize 50 into its prime factors, which is $2 \times 5^2$. Since there is no perfect square factor, we cannot simplify the square root of 50.

Conclusion

In conclusion, simplifying square roots is an essential skill in mathematics that has numerous real-world applications. By breaking down the process step by step and identifying the perfect square factor, we can simplify the square root of a number. In this article, we answered some frequently asked questions related to simplifying square roots and provided example problems to illustrate the concept.

Frequently Asked Questions

• Q: What is the square root of 48? A: The square root of 48 is $4\sqrt{3}$.
• Q: How do I simplify the square root of a number? A: To simplify the square root of a number, you need to factorize the number into its prime factors and identify the perfect square factor.
• Q: What is the perfect square factor? A: The perfect square factor is a factor that can be expressed as the square of an integer.

Further Reading

If you want to learn more about simplifying square roots, I recommend checking out the following resources:

• Khan Academy: Simplifying Square Roots
• Mathway: Simplifying Square Roots
• Wolfram Alpha: Simplifying Square Roots

References

• "Algebra" by Michael Artin
• "Geometry" by Michael Spivak
• "Calculus" by Michael Spivak