Evaluate The Expression Without Using A Calculator: 16 − 1 4 16^{-\frac{1}{4}} 1 6 − 4 1 ​ A. -4 B. − 1 2 -\frac{1}{2} − 2 1 ​ C. 1 2 \frac{1}{2} 2 1 ​ D. -2

Introduction

In this article, we will evaluate the expression $16^{-\frac{1}{4}}$ without using a calculator. This involves understanding the properties of exponents and how to simplify expressions with negative exponents. We will also explore the concept of fractional exponents and how they can be used to simplify expressions.

Understanding Exponents

Exponents are a shorthand way of writing repeated multiplication. For example, $2^3$ can be written as $2 \times 2 \times 2$. Exponents can also be used to represent negative numbers, such as $2^{-3}$, which can be written as $\frac{1}{2^3}$.

Simplifying Expressions with Negative Exponents

When we have an expression with a negative exponent, such as $16^{-\frac{1}{4}}$, we can simplify it by taking the reciprocal of the base and changing the sign of the exponent. In this case, we can rewrite the expression as $\frac{1}{16^{\frac{1}{4}}}$.

Understanding Fractional Exponents

Fractional exponents are a way of expressing roots. For example, $a^{\frac{1}{2}}$ can be written as $\sqrt{a}$. Similarly, $a^{\frac{1}{4}}$ can be written as $\sqrt[4]{a}$.

Simplifying the Expression

Now that we have simplified the expression to $\frac{1}{16^{\frac{1}{4}}}$, we can further simplify it by finding the fourth root of 16. The fourth root of 16 is 2, since $2^4 = 16$. Therefore, we can rewrite the expression as $\frac{1}{2}$.

Conclusion

In conclusion, we have evaluated the expression $16^{-\frac{1}{4}}$ without using a calculator. We simplified the expression by taking the reciprocal of the base and changing the sign of the exponent, and then further simplified it by finding the fourth root of 16. The final answer is $\frac{1}{2}$.

Final Answer

The final answer is $\boxed{\frac{1}{2}}$.

Discussion

The expression $16^{-\frac{1}{4}}$ can be evaluated without using a calculator by understanding the properties of exponents and how to simplify expressions with negative exponents. The key concept here is to take the reciprocal of the base and change the sign of the exponent, and then to find the fourth root of 16. This can be a useful technique for evaluating expressions with fractional exponents.

Related Topics

• Evaluating expressions with negative exponents
• Simplifying expressions with fractional exponents
• Understanding the properties of exponents

Example Problems

• Evaluate the expression $25^{-\frac{1}{2}}$ without using a calculator.
• Simplify the expression $9^{\frac{1}{3}}$.
• Evaluate the expression $4^{-\frac{1}{3}}$ without using a calculator.

Solutions

• The expression $25^{-\frac{1}{2}}$ can be evaluated as $\frac{1}{5}$.
• The expression $9^{\frac{1}{3}}$ can be simplified as $\sqrt[3]{9}$.
• The expression $4^{-\frac{1}{3}}$ can be evaluated as $\frac{1}{\sqrt[3]{4}}$.

Conclusion

In conclusion, we have evaluated the expression $16^{-\frac{1}{4}}$ without using a calculator. We simplified the expression by taking the reciprocal of the base and changing the sign of the exponent, and then further simplified it by finding the fourth root of 16. The final answer is $\frac{1}{2}$.

Introduction

In our previous article, we evaluated the expression $16^{-\frac{1}{4}}$ without using a calculator. We simplified the expression by taking the reciprocal of the base and changing the sign of the exponent, and then further simplified it by finding the fourth root of 16. In this article, we will answer some common questions related to evaluating expressions without using a calculator.

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

A: A negative exponent is a shorthand way of writing the reciprocal of the base with a positive exponent. For example, $2^{-3}$ can be written as $\frac{1}{2^3}$. A positive exponent, on the other hand, represents repeated multiplication. For example, $2^3$ can be written as $2 \times 2 \times 2$.

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

A: To simplify an expression with a negative exponent, take the reciprocal of the base and change the sign of the exponent. For example, $16^{-\frac{1}{4}}$ can be simplified as $\frac{1}{16^{\frac{1}{4}}}$.

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

A: A fractional exponent represents a root, while a negative exponent represents the reciprocal of the base with a positive exponent. For example, $a^{\frac{1}{2}}$ can be written as $\sqrt{a}$, while $a^{-2}$ can be written as $\frac{1}{a^2}$.

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

A: To evaluate an expression with a fractional exponent, find the root of the base. For example, $16^{\frac{1}{4}}$ can be evaluated as $\sqrt[4]{16}$, which is equal to 2.

Q: Can I use a calculator to evaluate expressions with exponents?

A: While it is possible to use a calculator to evaluate expressions with exponents, it is not always necessary. By understanding the properties of exponents and how to simplify expressions with negative and fractional exponents, you can evaluate expressions without using a calculator.

Q: What are some common mistakes to avoid when evaluating expressions with exponents?

A: Some common mistakes to avoid when evaluating expressions with exponents include:

• Not simplifying expressions with negative exponents
• Not understanding the difference between a fractional exponent and a negative exponent
• Not finding the root of the base when evaluating an expression with a fractional exponent
• Not using the correct order of operations when evaluating expressions with multiple exponents

Q: How can I practice evaluating expressions with exponents?

A: You can practice evaluating expressions with exponents by working through example problems and exercises. You can also try simplifying expressions with negative and fractional exponents to help you understand the concepts better.

Q: What are some real-world applications of evaluating expressions with exponents?

A: Evaluating expressions with exponents has many real-world applications, including:

• Calculating interest rates and investments
• Determining the area and volume of shapes
• Evaluating scientific data and formulas
• Solving problems in physics and engineering

Conclusion

In conclusion, evaluating expressions with exponents is an important skill that can be used in a variety of real-world applications. By understanding the properties of exponents and how to simplify expressions with negative and fractional exponents, you can evaluate expressions without using a calculator. We hope that this Q&A article has helped you to better understand the concepts and has provided you with the tools and resources you need to practice and apply your skills.