Evaluate The Following. Click On Not A Real Number If Applicable.(a) − 32 1 5 = -32^{\frac{1}{5}}= − 3 2 5 1 = (b) − 81 1 4 = -81^{\frac{1}{4}}= − 8 1 4 1 =

Mar 12, 2025 by ADMIN 163 views

Evaluating Expressions with Negative Bases and Fractional Exponents

In mathematics, expressions with negative bases and fractional exponents can be challenging to evaluate. The correct evaluation of such expressions requires a thorough understanding of the properties of exponents and the behavior of negative numbers. In this article, we will evaluate two expressions: $-32^{\frac{1}{5}}$ and $-81^{\frac{1}{4}}$ . We will also discuss the concept of "not a real number" and its implications in mathematics.

Understanding Negative Bases and Fractional Exponents

A negative base is a number that is less than zero, while a fractional exponent is a number that is expressed as a fraction. When we combine these two concepts, we get expressions that can be challenging to evaluate. For example, $-32^{\frac{1}{5}}$ involves a negative base ( $-32$ ) and a fractional exponent ( $\frac{1}{5}$ ).

To evaluate such expressions, we need to understand the properties of exponents. One of the key properties of exponents is that when we raise a number to a power, we can multiply the number by itself as many times as the exponent indicates. For example, $a^m \cdot a^n = a^{m+n}$ . This property can be extended to negative bases and fractional exponents.

Evaluating $-32^{\frac{1}{5}}$

To evaluate $-32^{\frac{1}{5}}$ , we need to understand that the expression involves a negative base ( $-32$ ) and a fractional exponent ( $\frac{1}{5}$ ). We can start by rewriting the expression as $(-32)^{\frac{1}{5}}$ . This is because the negative sign can be moved to the other side of the exponent.

Next, we can use the property of exponents that states $a^m \cdot a^n = a^{m+n}$ . In this case, we can rewrite $(-32)^{\frac{1}{5}}$ as $(-32)^{\frac{1}{5}} \cdot (-32)^0$ . This is because $(-32)^0 = 1$ , and we can multiply any number by 1 without changing its value.

Now, we can simplify the expression by combining the two terms. We get $(-32)^{\frac{1}{5}} \cdot (-32)^0 = (-32)^{\frac{1}{5} + 0} = (-32)^{\frac{1}{5}}$ .

However, we still need to evaluate the expression $(-32)^{\frac{1}{5}}$ . To do this, we can use the fact that $(-32)^{\frac{1}{5}} = \sqrt[5]{-32}$ . This is because the fractional exponent $\frac{1}{5}$ indicates that we need to take the fifth root of the number.

The fifth root of $-32$ is a number that, when raised to the power of 5, gives us $-32$ . This number is $-2$ , because $(-2)^5 = -32$ .

Therefore, the value of $-32^{\frac{1}{5}}$ is $-2$ .

Evaluating $-81^{\frac{1}{4}}$

To evaluate $-81^{\frac{1}{4}}$ , we can follow a similar process to the one we used to evaluate $-32^{\frac{1}{5}}$ . We can start by rewriting the expression as $(-81)^{\frac{1}{4}}$ .

Next, we can use the property of exponents that states $a^m \cdot a^n = a^{m+n}$ . In this case, we can rewrite $(-81)^{\frac{1}{4}}$ as $(-81)^{\frac{1}{4}} \cdot (-81)^0$ . This is because $(-81)^0 = 1$ , and we can multiply any number by 1 without changing its value.

Now, we can simplify the expression by combining the two terms. We get $(-81)^{\frac{1}{4}} \cdot (-81)^0 = (-81)^{\frac{1}{4} + 0} = (-81)^{\frac{1}{4}}$ .

However, we still need to evaluate the expression $(-81)^{\frac{1}{4}}$ . To do this, we can use the fact that $(-81)^{\frac{1}{4}} = \sqrt[4]{-81}$ . This is because the fractional exponent $\frac{1}{4}$ indicates that we need to take the fourth root of the number.

The fourth root of $-81$ is a number that, when raised to the power of 4, gives us $-81$ . This number is $-3$ , because $(-3)^4 = 81$ .

However, we need to consider the fact that $-81$ is a negative number, and the fourth root of a negative number is not a real number. This is because the fourth root of a negative number would require us to take the square root of a negative number, which is not possible in the real number system.

Therefore, the value of $-81^{\frac{1}{4}}$ is "not a real number".

Conclusion

In this article, we evaluated two expressions: $-32^{\frac{1}{5}}$ and $-81^{\frac{1}{4}}$ . We used the properties of exponents to simplify the expressions and evaluate them. We also discussed the concept of "not a real number" and its implications in mathematics.

The value of $-32^{\frac{1}{5}}$ is $-2$ , while the value of $-81^{\frac{1}{4}}$ is "not a real number". This highlights the importance of understanding the properties of exponents and the behavior of negative numbers in mathematics.

References

[1] "Exponents and Exponential Functions" by Math Open Reference
[2] "Roots of Negative Numbers" by Wolfram MathWorld
[3] "Properties of Exponents" by Khan Academy
Evaluating Expressions with Negative Bases and Fractional Exponents: Q&A

In our previous article, we evaluated two expressions: $-32^{\frac{1}{5}}$ and $-81^{\frac{1}{4}}$ . We used the properties of exponents to simplify the expressions and evaluate them. However, we may still have some questions about the process and the results. In this article, we will answer some of the most frequently asked questions about evaluating expressions with negative bases and fractional exponents.

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

A: A negative base is a number that is less than zero, while a positive base is a number that is greater than zero. For example, $-32$ is a negative base, while $32$ is a positive base.

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

A: To evaluate an expression with a negative base and a fractional exponent, you need to follow these steps:

Rewrite the expression with a positive base by moving the negative sign to the other side of the exponent.
Use the property of exponents that states $a^m \cdot a^n = a^{m+n}$ to simplify the expression.
Evaluate the expression by taking the root of the number indicated by the fractional exponent.

Q: What is the value of $-32^{\frac{1}{5}}$ ?

A: The value of $-32^{\frac{1}{5}}$ is $-2$ . This is because the fifth root of $-32$ is $-2$ , and $(-2)^5 = -32$ .

Q: What is the value of $-81^{\frac{1}{4}}$ ?

A: The value of $-81^{\frac{1}{4}}$ is "not a real number". This is because the fourth root of $-81$ is not a real number, as it would require us to take the square root of a negative number.

Q: Why is the fourth root of a negative number not a real number?

A: The fourth root of a negative number is not a real number because it would require us to take the square root of a negative number. In the real number system, we cannot take the square root of a negative number, as it would result in an imaginary number.

Q: What is the difference between a real number and an imaginary number?

A: A real number is a number that can be expressed as a decimal or a fraction, while an imaginary number is a number that cannot be expressed as a decimal or a fraction. Imaginary numbers are used to represent quantities that cannot be expressed in the real number system.

Q: Can I use imaginary numbers to evaluate expressions with negative bases and fractional exponents?

A: Yes, you can use imaginary numbers to evaluate expressions with negative bases and fractional exponents. However, you need to be careful when working with imaginary numbers, as they can be complex and difficult to work with.

Q: What are some common mistakes to avoid when evaluating expressions with negative bases and fractional exponents?

A: Some common mistakes to avoid when evaluating expressions with negative bases and fractional exponents include:

Not rewriting the expression with a positive base
Not using the property of exponents that states $a^m \cdot a^n = a^{m+n}$
Not evaluating the expression by taking the root of the number indicated by the fractional exponent
Not considering the possibility of imaginary numbers when working with negative bases and fractional exponents

Conclusion

In this article, we answered some of the most frequently asked questions about evaluating expressions with negative bases and fractional exponents. We discussed the properties of exponents, the behavior of negative numbers, and the use of imaginary numbers. We also highlighted some common mistakes to avoid when working with negative bases and fractional exponents. By following the steps outlined in this article, you can evaluate expressions with negative bases and fractional exponents with confidence.

References

[1] "Exponents and Exponential Functions" by Math Open Reference
[2] "Roots of Negative Numbers" by Wolfram MathWorld
[3] "Properties of Exponents" by Khan Academy
[4] "Imaginary Numbers" by Math Is Fun