What Is The Principal Square Root Of 25 \sqrt{25} 25 ?A. 5 B. 25 C. 50 D. 125

Mar 10, 2025 by ADMIN 83 views

**Understanding the Principal Square Root**

The principal square root of a number is a value that, when multiplied by itself, gives the original number. In mathematical terms, if we have a number $x$ , its principal square root is denoted by $\sqrt{x}$ and is a value $y$ such that $y^2 = x$ . The principal square root is also known as the positive square root.

What is the Principal Square Root of $\sqrt{25}$ ?

To find the principal square root of $\sqrt{25}$ , we need to understand that the expression $\sqrt{25}$ is already a square root. When we see $\sqrt{25}$ , it means that we are looking for a value that, when multiplied by itself, gives 25. In other words, we are looking for a number $x$ such that $x^2 = 25$ .

Solving for the Principal Square Root

To solve for the principal square root of $\sqrt{25}$ , we can start by finding the square root of 25. We know that $5^2 = 25$ , so the square root of 25 is 5. However, we need to find the principal square root, which is the positive square root.

Why is 5 the Principal Square Root of $\sqrt{25}$ ?

The reason why 5 is the principal square root of $\sqrt{25}$ is that it is the positive value that, when multiplied by itself, gives 25. In other words, $5^2 = 25$ , which means that 5 is the value that satisfies the equation $x^2 = 25$ . Since we are looking for the principal square root, which is the positive square root, we take the positive value, which is 5.

Why are the Other Options Incorrect?

Now, let's look at the other options and see why they are incorrect.

Option B: 25. The square root of 25 is 5, not 25. Therefore, 25 is not the principal square root of $\sqrt{25}$ .
Option C: 50. The square of 50 is 2500, not 25. Therefore, 50 is not the principal square root of $\sqrt{25}$ .
Option D: 125. The square of 125 is 15625, not 25. Therefore, 125 is not the principal square root of $\sqrt{25}$ .

Conclusion

In conclusion, the principal square root of $\sqrt{25}$ is 5. This is because 5 is the positive value that, when multiplied by itself, gives 25. The other options are incorrect because they do not satisfy the equation $x^2 = 25$ .

Key Takeaways

The principal square root of a number is a value that, when multiplied by itself, gives the original number.
The principal square root is also known as the positive square root.
To find the principal square root of a number, we need to find the positive value that, when multiplied by itself, gives the original number.

Final Answer

Frequently Asked Questions

Q: What is the principal square root of a number?

A: The principal square root of a number is a value that, when multiplied by itself, gives the original number. In mathematical terms, if we have a number $x$ , its principal square root is denoted by $\sqrt{x}$ and is a value $y$ such that $y^2 = x$ . The principal square root is also known as the positive square root.

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

A: To find the principal square root of a number, we need to find the positive value that, when multiplied by itself, gives the original number. We can do this by finding the square root of the number and taking the positive value.

Q: What is the difference between the principal square root and the negative square root?

A: The principal square root is the positive square root of a number, while the negative square root is the negative value that, when multiplied by itself, gives the original number. For example, the principal square root of 25 is 5, while the negative square root of 25 is -5.

Q: Can a number have more than one principal square root?

A: No, a number can only have one principal square root. The principal square root is the positive value that, when multiplied by itself, gives the original number.

Q: How do I know if a number has a principal square root?

A: Any positive number has a principal square root. For example, the principal square root of 25 is 5, while the principal square root of 36 is 6.

Q: Can a number have a principal square root that is not an integer?

A: Yes, a number can have a principal square root that is not an integer. For example, the principal square root of 2 is approximately 1.414, while the principal square root of 3 is approximately 1.732.

Q: How do I find the principal square root of a decimal number?

A: To find the principal square root of a decimal number, we can use a calculator or a computer program to find the square root of the number and take the positive value.

Q: Can a number have a principal square root that is a fraction?

A: Yes, a number can have a principal square root that is a fraction. For example, the principal square root of 2/9 is approximately 0.471, while the principal square root of 3/4 is approximately 0.866.

Q: How do I know if a number has a principal square root that is a fraction?

A: We can find the principal square root of a fraction by finding the square root of the numerator and the denominator separately and then taking the positive value.

Q: Can a number have a principal square root that is a complex number?

A: Yes, a number can have a principal square root that is a complex number. For example, the principal square root of -1 is i, while the principal square root of -4 is 2i.

Q: How do I find the principal square root of a complex number?

A: To find the principal square root of a complex number, we can use the formula $\sqrt{a+bi} = \pm \left(\sqrt{\frac{\sqrt{a^2+b^2}+a}{2}} + i\sqrt{\frac{\sqrt{a^2+b^2}-a}{2}}\right)$ , where $a$ and $b$ are the real and imaginary parts of the complex number.

Q: Can a number have a principal square root that is a negative number?

A: No, a number cannot have a principal square root that is a negative number. The principal square root is always a positive value.

Q: How do I know if a number has a principal square root that is a negative number?

A: We can find the principal square root of a number by finding the square root of the number and taking the positive value. If the result is a negative number, then the number does not have a principal square root.

Q: Can a number have a principal square root that is an imaginary number?

A: No, a number cannot have a principal square root that is an imaginary number. The principal square root is always a real value.

Q: How do I know if a number has a principal square root that is an imaginary number?

A: We can find the principal square root of a number by finding the square root of the number and taking the positive value. If the result is an imaginary number, then the number does not have a principal square root.

Q: Can a number have a principal square root that is a real number and an imaginary number at the same time?

A: No, a number cannot have a principal square root that is a real number and an imaginary number at the same time. The principal square root is always a real value.

Q: How do I know if a number has a principal square root that is a real number and an imaginary number at the same time?

A: We can find the principal square root of a number by finding the square root of the number and taking the positive value. If the result is a real number and an imaginary number at the same time, then the number does not have a principal square root.

Q: Can a number have a principal square root that is a complex number and a real number at the same time?

A: No, a number cannot have a principal square root that is a complex number and a real number at the same time. The principal square root is always a real value.

Q: How do I know if a number has a principal square root that is a complex number and a real number at the same time?

A: We can find the principal square root of a number by finding the square root of the number and taking the positive value. If the result is a complex number and a real number at the same time, then the number does not have a principal square root.

Q: Can a number have a principal square root that is a real number and a complex number at the same time?

A: No, a number cannot have a principal square root that is a real number and a complex number at the same time. The principal square root is always a real value.

Q: How do I know if a number has a principal square root that is a real number and a complex number at the same time?

A: We can find the principal square root of a number by finding the square root of the number and taking the positive value. If the result is a real number and a complex number at the same time, then the number does not have a principal square root.

Q: Can a number have a principal square root that is a complex number and an imaginary number at the same time?

A: No, a number cannot have a principal square root that is a complex number and an imaginary number at the same time. The principal square root is always a real value.

Q: How do I know if a number has a principal square root that is a complex number and an imaginary number at the same time?

A: We can find the principal square root of a number by finding the square root of the number and taking the positive value. If the result is a complex number and an imaginary number at the same time, then the number does not have a principal square root.