What Is The Following Product? Assume B ≥ 0 B \geq 0 B ≥ 0 . B ⋅ B \sqrt{b} \cdot \sqrt{b} B ​ ⋅ B ​ A. B B B \sqrt{b} B B ​ B. 2 B 2 \sqrt{b} 2 B ​ C. B B B D. B 2 B^2 B 2

by ADMIN 173 views

Understanding the Problem

The given problem involves a mathematical expression that requires simplification. We are asked to find the product of two square roots, denoted as b⋅b\sqrt{b} \cdot \sqrt{b}. To solve this problem, we need to apply the properties of square roots and simplify the expression.

Properties of Square Roots

Before we proceed with the simplification, let's recall the properties of square roots. The square root of a number bb is denoted as b\sqrt{b} and is defined as the number that, when multiplied by itself, gives the original number bb. In other words, b⋅b=b\sqrt{b} \cdot \sqrt{b} = b.

Simplifying the Expression

Now, let's apply the properties of square roots to simplify the given expression. We have b⋅b\sqrt{b} \cdot \sqrt{b}. Using the property mentioned above, we can rewrite this expression as:

b⋅b=b⋅b\sqrt{b} \cdot \sqrt{b} = \sqrt{b \cdot b}

Applying the Property of Square Roots

Since the square root of a product is equal to the product of the square roots, we can simplify the expression further:

b⋅b=b2\sqrt{b \cdot b} = \sqrt{b^2}

Evaluating the Square Root

Now, let's evaluate the square root of b2b^2. Since the square root of a number is the number that, when multiplied by itself, gives the original number, we can rewrite the square root of b2b^2 as:

b2=b\sqrt{b^2} = b

Conclusion

In conclusion, the product of two square roots, denoted as b⋅b\sqrt{b} \cdot \sqrt{b}, is equal to bb. Therefore, the correct answer is:

The Final Answer is: C. bb

Why is this the Correct Answer?

The correct answer is C. bb because the product of two square roots is equal to the number inside the square root, which is bb. This is a fundamental property of square roots, and it is essential to understand this property to solve problems involving square roots.

What is the Significance of this Problem?

This problem may seem simple, but it is essential to understand the properties of square roots to solve more complex problems in mathematics. The concept of square roots is used extensively in various fields, including algebra, geometry, and calculus. Therefore, it is crucial to have a solid understanding of this concept to succeed in these fields.

Real-World Applications

The concept of square roots has numerous real-world applications. For example, in physics, the square root of a number is used to calculate the speed of an object. In engineering, the square root of a number is used to calculate the stress on a material. In finance, the square root of a number is used to calculate the volatility of a stock.

Common Mistakes to Avoid

When simplifying expressions involving square roots, it is essential to remember the following common mistakes to avoid:

  • Not applying the property of square roots correctly
  • Not simplifying the expression correctly
  • Not evaluating the square root correctly

Tips and Tricks

To simplify expressions involving square roots, follow these tips and tricks:

  • Always apply the property of square roots correctly
  • Simplify the expression step by step
  • Evaluate the square root correctly

Conclusion

In conclusion, the product of two square roots, denoted as b⋅b\sqrt{b} \cdot \sqrt{b}, is equal to bb. This is a fundamental property of square roots, and it is essential to understand this property to solve problems involving square roots. The concept of square roots has numerous real-world applications, and it is crucial to have a solid understanding of this concept to succeed in various fields.

Final Thoughts

The concept of square roots is a fundamental concept in mathematics, and it is essential to understand this concept to solve problems involving square roots. The product of two square roots is equal to the number inside the square root, which is bb. This is a property that is used extensively in various fields, including algebra, geometry, and calculus. Therefore, it is crucial to have a solid understanding of this concept to succeed in these fields.

References

Q: What is the product of two square roots?

A: The product of two square roots is equal to the number inside the square root. In other words, if we have b⋅b\sqrt{b} \cdot \sqrt{b}, the product is equal to bb.

Q: Why is this the case?

A: This is because the square root of a number is the number that, when multiplied by itself, gives the original number. Therefore, when we multiply two square roots together, we are essentially multiplying the number inside the square root by itself, which gives us the original number.

Q: Can you provide an example?

A: Let's say we have 4⋅4\sqrt{4} \cdot \sqrt{4}. Using the property mentioned above, we can simplify this expression as follows:

4⋅4=4⋅4=16=4\sqrt{4} \cdot \sqrt{4} = \sqrt{4 \cdot 4} = \sqrt{16} = 4

Q: What if the numbers inside the square roots are different?

A: If the numbers inside the square roots are different, we can still apply the property mentioned above. For example, let's say we have 2⋅3\sqrt{2} \cdot \sqrt{3}. Using the property, we can simplify this expression as follows:

2⋅3=2⋅3=6\sqrt{2} \cdot \sqrt{3} = \sqrt{2 \cdot 3} = \sqrt{6}

Q: Can you provide more examples?

A: Here are a few more examples:

  • 9⋅9=9⋅9=81=9\sqrt{9} \cdot \sqrt{9} = \sqrt{9 \cdot 9} = \sqrt{81} = 9
  • 16⋅25=16⋅25=400=20\sqrt{16} \cdot \sqrt{25} = \sqrt{16 \cdot 25} = \sqrt{400} = 20
  • 3⋅5=3⋅5=15\sqrt{3} \cdot \sqrt{5} = \sqrt{3 \cdot 5} = \sqrt{15}

Q: What if we have a negative number inside the square root?

A: If we have a negative number inside the square root, we need to be careful when simplifying the expression. For example, let's say we have −4⋅−4\sqrt{-4} \cdot \sqrt{-4}. Using the property mentioned above, we can simplify this expression as follows:

−4⋅−4=−4⋅−4=16=4\sqrt{-4} \cdot \sqrt{-4} = \sqrt{-4 \cdot -4} = \sqrt{16} = 4

However, we need to be careful when dealing with negative numbers inside the square root, as the square root of a negative number is not a real number.

Q: Can you provide more examples with negative numbers?

A: Here are a few more examples:

  • −9⋅−9=−9⋅−9=81=9\sqrt{-9} \cdot \sqrt{-9} = \sqrt{-9 \cdot -9} = \sqrt{81} = 9
  • −16⋅−25=−16⋅−25=400=20\sqrt{-16} \cdot \sqrt{-25} = \sqrt{-16 \cdot -25} = \sqrt{400} = 20
  • −3⋅−5=−3⋅−5=15\sqrt{-3} \cdot \sqrt{-5} = \sqrt{-3 \cdot -5} = \sqrt{15}

Q: What if we have a complex number inside the square root?

A: If we have a complex number inside the square root, we need to be careful when simplifying the expression. For example, let's say we have 3+4i⋅3+4i\sqrt{3+4i} \cdot \sqrt{3+4i}. Using the property mentioned above, we can simplify this expression as follows:

3+4i⋅3+4i=(3+4i)(3+4i)=9+24i+16i2=9+24i−16=−7+24i\sqrt{3+4i} \cdot \sqrt{3+4i} = \sqrt{(3+4i)(3+4i)} = \sqrt{9+24i+16i^2} = \sqrt{9+24i-16} = \sqrt{-7+24i}

However, we need to be careful when dealing with complex numbers inside the square root, as the square root of a complex number is not a real number.

Q: Can you provide more examples with complex numbers?

A: Here are a few more examples:

  • 2+3i⋅2+3i=(2+3i)(2+3i)=4+12i+9i2=4+12i−9=−5+12i\sqrt{2+3i} \cdot \sqrt{2+3i} = \sqrt{(2+3i)(2+3i)} = \sqrt{4+12i+9i^2} = \sqrt{4+12i-9} = \sqrt{-5+12i}
  • 4+5i⋅4+5i=(4+5i)(4+5i)=16+40i+25i2=16+40i−25=−9+40i\sqrt{4+5i} \cdot \sqrt{4+5i} = \sqrt{(4+5i)(4+5i)} = \sqrt{16+40i+25i^2} = \sqrt{16+40i-25} = \sqrt{-9+40i}
  • 3+2i⋅3+2i=(3+2i)(3+2i)=9+12i+4i2=9+12i−4=5+12i\sqrt{3+2i} \cdot \sqrt{3+2i} = \sqrt{(3+2i)(3+2i)} = \sqrt{9+12i+4i^2} = \sqrt{9+12i-4} = \sqrt{5+12i}

Conclusion

In conclusion, the product of two square roots is equal to the number inside the square root. This is a fundamental property of square roots, and it is essential to understand this property to solve problems involving square roots. We have provided several examples to illustrate this property, including examples with negative numbers and complex numbers. We hope this article has been helpful in understanding the product of two square roots.