What Is The Following Product? 10 ⋅ 10 \sqrt{10} \cdot \sqrt{10} 10 ​ ⋅ 10 ​ A. 10 B. 10 10 10 \sqrt{10} 10 10 ​ C. 100 D. 2 10 2 \sqrt{10} 2 10 ​

Understanding the Basics of Square Roots

In mathematics, 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. The square root of a number is denoted by the symbol √. In this case, we are given the expression $\sqrt{10} \cdot \sqrt{10}$, and we need to determine the value of this product.

The Product of Square Roots

When we multiply two square roots together, we can combine them into a single square root. This is because the square root of a number is the value that, when multiplied by itself, gives the original number. Therefore, when we multiply two square roots together, we are essentially multiplying the values inside the square roots together.

Applying the Rule of Square Roots

In this case, we have $\sqrt{10} \cdot \sqrt{10}$. Using the rule of square roots, we can combine these two square roots into a single square root. This gives us $\sqrt{10 \cdot 10}$.

Simplifying the Expression

Now, we can simplify the expression inside the square root. When we multiply 10 by 10, we get 100. Therefore, the expression becomes $\sqrt{100}$.

Evaluating the Square Root

The square root of 100 is 10, because 10 multiplied by 10 equals 100. Therefore, the value of the expression $\sqrt{10} \cdot \sqrt{10}$ is 10.

Conclusion

In conclusion, the value of the expression $\sqrt{10} \cdot \sqrt{10}$ is 10. This is because when we multiply two square roots together, we can combine them into a single square root, and the value inside the square root is the product of the values inside the original square roots.

Answer

The correct answer is A. 10.

Why the Other Options are Incorrect

The other options are incorrect because they do not follow the rule of square roots. Option B, $10 \sqrt{10}$, is incorrect because it does not combine the two square roots into a single square root. Option C, 100, is incorrect because it does not take into account the fact that we are multiplying two square roots together. Option D, $2 \sqrt{10}$, is incorrect because it does not follow the rule of square roots and also introduces an incorrect value.

Common Mistakes to Avoid

When working with square roots, it is easy to make mistakes. One common mistake is to forget to combine the square roots into a single square root. Another common mistake is to introduce incorrect values or to forget to simplify the expression inside the square root.

Tips for Solving Similar Problems

When solving similar problems, it is essential to remember the rule of square roots. This rule states that when we multiply two square roots together, we can combine them into a single square root. It is also essential to simplify the expression inside the square root and to evaluate the square root correctly.

Real-World Applications

The concept of square roots has many 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.

Conclusion

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Q: What is the difference between a square root and a square?

A: A square root 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. A square, on the other hand, is the result of multiplying a number by itself. For example, the square of 4 is 16, because 4 multiplied by 4 equals 16.

Q: How do I simplify a square root expression?

A: To simplify a square root expression, you need to find the largest perfect square that divides the number inside the square root. For example, the square root of 20 can be simplified as follows:

√20 = √(4 × 5) = √4 × √5 = 2√5

Q: What is the rule for multiplying square roots?

A: The rule for multiplying square roots is that you can combine them into a single square root. For example:

√a × √b = √(a × b)

Q: Can I simplify a square root expression by factoring?

A: Yes, you can simplify a square root expression by factoring. For example:

√(12 × 15) = √(4 × 3 × 5) = √(4 × 5) × √3 = 2√(3 × 5) = 2√15

Q: How do I evaluate a square root expression?

A: To evaluate a square root expression, you need to find the value of the number inside the square root. For example:

√16 = 4, because 4 multiplied by 4 equals 16.

Q: Can I use a calculator to evaluate a square root expression?

A: Yes, you can use a calculator to evaluate a square root expression. Most calculators have a square root button that you can press to find the value of a square root expression.

Q: What is the difference between a rational and an irrational number?

A: A rational number is a number that can be expressed as the ratio of two integers. For example, 3/4 is a rational number. An irrational number, on the other hand, is a number that cannot be expressed as the ratio of two integers. For example, the square root of 2 is an irrational number.

Q: Can I simplify a square root expression by rationalizing the denominator?

A: Yes, you can simplify a square root expression by rationalizing the denominator. For example:

√(2/3) = √(2) / √(3) = √(2) / √(3) × √(3) / √(3) = (√(2) × √(3)) / (√(3) × √(3)) = (√(6)) / 3

Q: How do I use square roots in real-world applications?

A: Square roots are used in many real-world applications, such as physics, engineering, and finance. 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.

Q: Can I use square roots to solve problems in other areas of mathematics?

A: Yes, you can use square roots to solve problems in other areas of mathematics, such as algebra and geometry. For example, in algebra, you can use square roots to solve quadratic equations. In geometry, you can use square roots to calculate the length of a diagonal in a rectangle.

Conclusion

In conclusion, square roots are an essential concept in mathematics that has many real-world applications. By understanding the basics of square roots, you can simplify complex expressions and solve problems in other areas of mathematics.