Evaluate: ( 4 25 ) 1 2 \left(\frac{4}{25}\right)^{\frac{1}{2}} ( 25 4 ) 2 1 A. 4 5 \frac{4}{5} 5 4 B. 4 2 \frac{4}{2} 2 4

Mar 9, 2025 by ADMIN 130 views

$Evaluate: $\left(\frac{4}{25}\right)^{\frac{1}{2}}$$

Introduction

In mathematics, exponents and roots are fundamental concepts that play a crucial role in solving various mathematical problems. The expression $\left(\frac{4}{25}\right)^{\frac{1}{2}}$ involves both exponentiation and root extraction. In this article, we will delve into the evaluation of this expression and explore the underlying mathematical principles.

Understanding Exponents and Roots

Before we proceed with the evaluation of the given expression, it is essential to understand the concepts of exponents and roots. An exponent is a small number that is raised to a power, indicating how many times a base number is multiplied by itself. For instance, $2^3$ means $2$ multiplied by itself $3$ times, resulting in $8$ . On the other hand, a root is the inverse operation of exponentiation, where a number is raised to a power to obtain the original value. For example, $\sqrt{16}$ means finding the number that, when multiplied by itself, gives $16$ , which is $4$ .

Evaluating the Expression

Now that we have a solid understanding of exponents and roots, let's focus on evaluating the expression $\left(\frac{4}{25}\right)^{\frac{1}{2}}$ . To do this, we need to apply the rules of exponentiation and root extraction.

The expression $\left(\frac{4}{25}\right)^{\frac{1}{2}}$ can be rewritten as $\sqrt{\frac{4}{25}}$ . This is because the exponent $\frac{1}{2}$ is equivalent to taking the square root of the fraction.

Simplifying the Expression

To simplify the expression, we can start by finding the square root of the numerator and the denominator separately. The square root of $4$ is $2$ , and the square root of $25$ is $5$ .

Therefore, $\sqrt{\frac{4}{25}}$ can be rewritten as $\frac{\sqrt{4}}{\sqrt{25}}$ , which simplifies to $\frac{2}{5}$ .

Conclusion

Final Answer

The final answer to the expression $\left(\frac{4}{25}\right)^{\frac{1}{2}}$ is $\boxed{\frac{2}{5}}$ .

Comparison with Options

Now that we have evaluated the expression, let's compare our result with the given options.

Option A: $\frac{4}{5}$

Option B: $\frac{4}{2}$

Our result: $\frac{2}{5}$

As we can see, our result $\frac{2}{5}$ does not match with either of the given options. This is because the options provided are incorrect, and our evaluation of the expression has led to a different result.

Importance of Evaluating Expressions

Evaluating expressions like $\left(\frac{4}{25}\right)^{\frac{1}{2}}$ is crucial in mathematics, as it helps us understand the underlying principles of exponents and roots. By applying these principles, we can solve a wide range of mathematical problems and make informed decisions in various fields.

Real-World Applications

The concept of exponents and roots has numerous real-world applications. For instance, in finance, exponents are used to calculate compound interest, while in physics, roots are used to calculate distances and velocities.

Tips for Evaluating Expressions

When evaluating expressions like $\left(\frac{4}{25}\right)^{\frac{1}{2}}$ , it is essential to follow these tips:

Understand the concepts of exponents and roots
Apply the rules of exponentiation and root extraction
Simplify the expression using the properties of square roots
Compare the result with the given options

By following these tips, you can evaluate expressions like $\left(\frac{4}{25}\right)^{\frac{1}{2}}$ with confidence and accuracy.

Conclusion

In conclusion, the expression $\left(\frac{4}{25}\right)^{\frac{1}{2}}$ evaluates to $\frac{2}{5}$ . This result is obtained by applying the rules of exponentiation and root extraction, and simplifying the expression using the properties of square roots. By understanding the concepts of exponents and roots, we can solve a wide range of mathematical problems and make informed decisions in various fields.

Introduction

In our previous article, we evaluated the expression $\left(\frac{4}{25}\right)^{\frac{1}{2}}$ and found that it simplifies to $\frac{2}{5}$ . However, we received several questions from readers regarding the evaluation of this expression. In this article, we will address some of the most frequently asked questions and provide additional insights into evaluating expressions with exponents and roots.

Q1: What is the difference between a square root and an exponent?

A1: A square root and an exponent are related but distinct concepts. A square root is the inverse operation of squaring a number, while an exponent is a small number that is raised to a power, indicating how many times a base number is multiplied by itself. For example, $\sqrt{16}$ means finding the number that, when multiplied by itself, gives $16$ , which is $4$ . On the other hand, $2^3$ means $2$ multiplied by itself $3$ times, resulting in $8$ .

Q2: How do I evaluate an expression with a negative exponent?

A2: Evaluating an expression with a negative exponent involves applying the rule that $a^{-n} = \frac{1}{a^n}$ . For example, $2^{-3}$ can be rewritten as $\frac{1}{2^3}$ , which simplifies to $\frac{1}{8}$ .

Q3: What is the difference between a rational exponent and a radical?

A3: A rational exponent and a radical are related but distinct concepts. A rational exponent is an exponent that is a fraction, such as $\frac{1}{2}$ or $\frac{3}{4}$ . A radical, on the other hand, is a symbol that represents the square root of a number, such as $\sqrt{16}$ or $\sqrt{25}$ . For example, $\left(\frac{4}{25}\right)^{\frac{1}{2}}$ can be rewritten as $\sqrt{\frac{4}{25}}$ , which simplifies to $\frac{2}{5}$ .

Q4: How do I evaluate an expression with a fractional exponent?

A4: Evaluating an expression with a fractional exponent involves applying the rule that $\left(\frac{a}{b}\right)^{\frac{m}{n}} = \left(\frac{a^n}{b^m}\right)^{\frac{1}{n}}$ . For example, $\left(\frac{4}{25}\right)^{\frac{1}{2}}$ can be rewritten as $\left(\frac{4^2}{25^2}\right)^{\frac{1}{2}}$ , which simplifies to $\frac{2}{5}$ .

Q5: What is the difference between a positive and negative exponent?

A5: A positive exponent and a negative exponent are related but distinct concepts. A positive exponent indicates that the base number is multiplied by itself a certain number of times, while a negative exponent indicates that the base number is divided by itself a certain number of times. For example, $2^3$ means $2$ multiplied by itself $3$ times, resulting in $8$ , while $2^{-3}$ means $2$ divided by itself $3$ times, resulting in $\frac{1}{8}$ .

Q6: How do I evaluate an expression with a mixed exponent?

A6: Evaluating an expression with a mixed exponent involves applying the rule that $a^{m+n} = a^m \cdot a^n$ . For example, $2^{3+2}$ can be rewritten as $2^3 \cdot 2^2$ , which simplifies to $16$ .

Conclusion

In conclusion, evaluating expressions with exponents and roots requires a solid understanding of the underlying mathematical principles. By applying the rules of exponentiation and root extraction, we can simplify complex expressions and arrive at accurate results. We hope that this Q&A article has provided additional insights into evaluating expressions with exponents and roots, and we encourage readers to continue exploring this fascinating topic.

Final Tips

Understand the concepts of exponents and roots
Apply the rules of exponentiation and root extraction
Simplify the expression using the properties of square roots
Compare the result with the given options
Practice evaluating expressions with exponents and roots to build your confidence and accuracy.

By following these tips, you can become proficient in evaluating expressions with exponents and roots and make informed decisions in various fields.