Which Expression Is Equivalent To $\sqrt{100}$?A. 10 B. $10^2$ C. 50 D. $50^2$

When dealing with mathematical expressions, it's essential to understand the concepts of square roots and exponents. A square root of a number is a value that, when multiplied by itself, gives the original number. On the other hand, an exponent is a number that represents the power to which a base number is raised. In this article, we will explore which expression is equivalent to $\sqrt{100}$.

What is a Square Root?

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 can be represented by the symbol $\sqrt{}$. In the case of $\sqrt{100}$, we are looking for a value that, when multiplied by itself, gives 100.

What is an Exponent?

An exponent is a number that represents the power to which a base number is raised. For example, $2^3$ means 2 raised to the power of 3, which equals 8. Exponents can be used to simplify complex expressions and make them easier to work with. In the case of $10^2$, the exponent 2 represents the power to which the base number 10 is raised.

Evaluating the Options

Now that we have a basic understanding of square roots and exponents, let's evaluate the options given in the problem.

Option A: 10

The square root of 100 is a value that, when multiplied by itself, gives 100. Since 10 multiplied by 10 equals 100, option A is a possible solution.

Option B: $10^2$

The expression $10^2$ represents 10 raised to the power of 2, which equals 100. This option is also a possible solution.

Option C: 50

The square root of 100 is a value that, when multiplied by itself, gives 100. Since 50 multiplied by 50 equals 2500, option C is not a possible solution.

Option D: $50^2$

The expression $50^2$ represents 50 raised to the power of 2, which equals 2500. This option is also not a possible solution.

Conclusion

In conclusion, the expression equivalent to $\sqrt{100}$ is both option A and option B. Both 10 and $10^2$ represent the square root of 100. Therefore, the correct answer is both A and B.

Additional Examples

To further illustrate the concept of square roots and exponents, let's consider a few additional examples.

Example 1: $\sqrt{25}$

The square root of 25 is a value that, when multiplied by itself, gives 25. Since 5 multiplied by 5 equals 25, the square root of 25 is 5.

Example 2: $3^2$

The expression $3^2$ represents 3 raised to the power of 2, which equals 9. This is an example of an exponent being used to simplify a complex expression.

Example 3: $\sqrt{36}$

The square root of 36 is a value that, when multiplied by itself, gives 36. Since 6 multiplied by 6 equals 36, the square root of 36 is 6.

Real-World Applications

Understanding square roots and exponents has numerous real-world applications. For example, in physics, the square root of a number can be used to calculate the speed of an object. In finance, exponents can be used to calculate compound interest.

Conclusion

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In the previous article, we explored the concepts of square roots and exponents. In this article, we will answer some frequently asked questions related to these topics.

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

A: A square root of a number is a value that, when multiplied by itself, gives the original number. On the other hand, an exponent is a number that represents the power to which a base number is raised.

Q: How do I simplify a square root expression?

A: To simplify a square root expression, you can look for perfect squares that are factors of the number inside the square root. For example, $\sqrt{16}$ can be simplified to 4, because 4 multiplied by 4 equals 16.

Q: What is the relationship between square roots and exponents?

A: The square root of a number can be represented as an exponent. For example, $\sqrt{16}$ can be written as $4^2$, because 4 multiplied by 4 equals 16.

Q: How do I evaluate an expression with a square root and an exponent?

A: To evaluate an expression with a square root and an exponent, you need to follow the order of operations (PEMDAS). For example, $\sqrt{16^2}$ can be evaluated as follows:

1. Evaluate the exponent: $16^2 = 256$
2. Take the square root: $\sqrt{256} = 16$

Q: What is the difference between a positive and negative exponent?

A: A positive exponent represents a power to which a base number is raised. For example, $2^3$ means 2 raised to the power of 3. A negative exponent represents a reciprocal of a power to which a base number is raised. For example, $2^{-3}$ means 1 divided by 2 raised to the power of 3.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you can rewrite it as a positive exponent. For example, $2^{-3}$ can be rewritten as $\frac{1}{2^3}$, because 1 divided by 2 raised to the power of 3 equals 1 divided by 8.

Q: What is the relationship between square roots and logarithms?

A: The square root of a number can be represented as a logarithm. For example, $\sqrt{16}$ can be written as $\log_{4}{16}$, because 4 raised to the power of 2 equals 16.

Q: How do I evaluate an expression with a square root and a logarithm?

A: To evaluate an expression with a square root and a logarithm, you need to follow the order of operations (PEMDAS). For example, $\sqrt{\log_{4}{16}}$ can be evaluated as follows:

1. Evaluate the logarithm: $\log_{4}{16} = 2$
2. Take the square root: $\sqrt{2}$

Conclusion

In conclusion, understanding square roots and exponents is essential in mathematics. By following the order of operations and simplifying expressions, you can evaluate complex mathematical expressions with ease.