Problem 60 - College Algebra with Modeling and Visualization

Question

Evaluate the expression by band. Approximate the answer to the nearest hundredth when appropriate. $$ (-32)^{1 / 5} $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Expression

The expression (-32)^{1/5} involves taking the fifth root of a negative number, which is mathematically possible because any real negative number raised to an odd-root is also a real number.

Step 2: Applying the Fifth Root

To solve (-32)^{1/5}, we find the number which raised to the fifth power equals -32. Given that the fifth root is defined as raising a number to the 1/5 power, this suggests finding a number x such that x^5 = -32.

Step 3: Evaluating the Fifth Root

The number -2 raised to the fifth power gives -32 because (-2)^5 = (-2) imes (-2) imes (-2) imes (-2) imes (-2) = -32. Therefore, (-32)^{1/5}=-2.

Step 4: Conclusion

The real number x that satisfies the equation is -2. Hence, (-32)^{1/5} = -2.

Key Concepts

Rational Exponents

Rational exponents are an intriguing concept in algebra, where exponents are expressed as fractions rather than whole numbers. Essentially, a rational exponent such as \( a^{m/n} \) is equivalent to the \( n \)-th root of \( a \) raised to the power of \( m \). This means that the exponent \( 1/n \) signifies finding the \( n \)-th root of the number, and then raising it to the power of \( m \) if \( m > 1 \).

Let's break it down further:

• \( a^{1/n} \) means the \( n \)-th root of \( a \).
• \( a^{m/n} \) means \( (a^{1/n})^m \), thus first taking the root and then raising it to the power of \( m \).

This expression allows us to extend the idea of exponents beyond integers, giving us more flexibility when dealing with roots and powers.
In the original exercise, \((-32)^{1/5}\) uses a rational exponent to denote the fifth root of \(-32\). This highlights how rational exponents simplify the notation when dealing with roots, making calculations concise and streamlined.

Odd Roots

Odd roots of numbers, especially negative ones, hold particular interest because they result in another real number. When finding an odd root, such as the cube root or fifth root, the process differs from even roots. For instance, while a square root of a negative number leads to complex numbers, an odd root remains within the realm of real numbers.

The reason behind this is simple: an odd number of negative multiplications results in a negative number.
Consider some critical points related to odd roots:

• The cube root of \(-8\) is \(-2\) because \((-2)^3 = -8\).
• The fifth root of \(-32\), as in the original exercise, is \(-2\) since \((-2)^5 = -32\).

Understanding odd roots offers a foundational understanding crucial for solving more complex algebraic problems that involve negative numbers and rational exponents.

Negative Numbers

Negative numbers are numbers that are less than zero, and they appear frequently in algebraic expressions. When it comes to taking roots, negative numbers behave differently depending on whether the roots are odd or even.

Key aspects of handling negative numbers include:

• Odd roots of negative numbers result in negative real numbers. This explains why \((-32)^{1/5} = -2\).
• Even roots of negative numbers do not produce real numbers, which means calculations involving them result in complex numbers.

In working with algebraic expressions, understanding how negative numbers interact with other concepts like exponents and roots is essential. Grasping these interactions aids in solving equations where the presence of negative signs influences the results but allows the continuation within the domain of real numbers for odd instances.

Real Numbers

Real numbers encompass a wide range of possibilities, including both rational and irrational numbers. They include all whole numbers, integers, fractions, and decimal numbers that can be pinpointed on a number line. The set of real numbers is extensive, covering both positive and negative values, along with zero.

Key points about real numbers include:

• Every point on the real number line represents a real number.
• The real number system includes subsets like natural numbers, whole numbers, and integers.
• Operations on real numbers such as addition, subtraction, multiplication, and division (except by zero) always result in another real number.

In algebra, expressions involving real numbers can include a wide range of calculations. The result of \((-32)^{1/5}\) being \(-2\) shows how real numbers continue to hold even when negative values and odd roots are involved. This flexibility makes them useful in various areas of mathematics, helping to solve numerous real-world problems.