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Chapter 10: Problem 22

Evaluate each exponential. $$ (-32)^{1 / 5} $$

Short Answer

Expert verified
(-32)^{1/5} = -2

Step by step solution

01

Understanding the Expression

The expression to evaluate is (-32)^{1/5}. This means we need to find the 5th root of -32. The exponent 1/5 represents the fifth root.
02

Calculating the Fifth Root

To find the fifth root of -32, we need to find a number that, when raised to the power of 5, equals -32. In this case, the number is -2 because (-2)^5 = -32.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

fifth root
When we talk about a fifth root, we're referring to a number which, when multiplied by itself four more times (so, five times in total), gives the original number. For instance, the fifth root of 32 is 2 because when you multiply 2 five times (2脳2脳2脳2脳2), you get 32. Similarly, (-2) is the fifth root of -32 because when multiplied by itself five times (鈭�2脳鈭�2脳鈭�2脳鈭�2脳鈭�2), you get -32.
negative base
A negative base is simply a number that is below zero, for instance, -2. When you raise a negative base to an odd exponent, such as 3 or 5, the result remains negative. This is because a negative number multiplied an odd number of times remains negative. For example, if you raise -2 to the power of 5, the calculation (-2)^5 equals -32, the negative value of the base. On the other hand, if the exponent is even, the result becomes positive. For instance, (-2)^4 results in 16.
exponential notation
Exponential notation is a shorthand way of expressing numbers by using a base and an exponent. The base is the number that is being multiplied, and the exponent tells you how many times to use the base in the multiplication. In the expression (-32)^{1/5}, -32 is the base, and 1/5 is the exponent. The exponent 1/5 means we are looking for the fifth root of the base, which is another way of writing in exponential notation. This simplifies calculations and makes it easier to write out and read large or complex numbers.

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