Problem 14 - College Algebra with Modeling and Visualization

Question

Evaluate the expression by hand. $$ (-32)^{-3 / 5} $$

Step-by-Step Solution

Answer

The value of \((-32)^{-3/5}\) is \(-\frac{1}{8}\).

Step 1: Understand the Problem

The expression \[ (-32)^{-3/5} \] requires us to find the reciprocal of \[ (-32)^{3/5} \] since the exponent is negative. Recall that a negative exponent \( a^{-b} \) is equivalent to \( \frac{1}{a^b} \).

Step 2: Rewrite the Expression

Rewrite the problem as the reciprocal of the positive exponent:\[ (-32)^{-3/5} = \frac{1}{(-32)^{3/5}} \]

Step 3: Break Down the Exponent

The exponent \( \frac{3}{5} \) can be interpreted as a power of a root. Interpret it as finding the 5th root and then cubing the result. In other words:\[ (-32)^{3/5} = \left((-32)^{1/5}\right)^3 \].

Step 4: Evaluate the 5th Root

Find the fifth root of \(-32\). Since \((-2)^5 = -32\), we conclude \((-32)^{1/5} = -2\).

Step 5: Calculate the Cube

Now that you have \((-32)^{1/5} = -2\), calculate \((-2)^3\):\((-2)^3 = -8\). This means \((-32)^{3/5} = -8\).

Step 6: Final Evaluation

Now compute the reciprocal as per Step 1:\[ (-32)^{-3/5} = \frac{1}{(-32)^{3/5}} = \frac{1}{-8} = -\frac{1}{8} \].

Key Concepts

Negative Exponents

Negative exponents might seem tricky at first, but they follow a simple rule. When you have any number (let's call it \(a\)) raised to a negative power \(-b\), it means you're dealing with the reciprocal of the positive exponent. In mathematical terms, \(a^{-b} = \frac{1}{a^b}\).
This helps to transform what might be a complex expression into something more manageable. Sometimes, students find it confusing to think about the negative sign; always remember that it just means "take the reciprocal."
For example, in the exercise \((-32)^{-3/5}\), we first consider it with a positive exponent: the expression becomes \(\frac{1}{(-32)^{3/5}}\). Understanding negative exponents is crucial in algebra, as it streamlines calculations involving powers.

Fractional Exponents

Fractional exponents express both roots and powers of a number in a concise form. For instance, if a number is raised to the power of \(\frac{m}{n}\), this represents taking the \(n\)-th root of the number and then raising the result to the \(m\)-th power. You can remember this as a two-step process:

• Step 1: Find the \(n\)-th root (\(a^{1/n}\))
• Step 2: Raise the result to the \(m\)-th power ((\(a^{1/n})^m\))

This simplifies complicated calculations, such as finding \((-32)^{3/5}\) as shown in the exercise. First, you find the 5th root of \(-32\) (which is \(-2\)), and then you cube the result. This breakdown makes it clear and manageable.
Recognizing how fractional exponents work leads to quicker and more efficient problem-solving in algebra and calculus.

Reciprocals

A reciprocal is simply turning a number upside down. In mathematical terms, if you have a number \(a\), its reciprocal is \(\frac{1}{a}\). This is a fundamental operation in math, especially when dealing with fractions, division, and negative exponents.
Reciprocals come from the idea of multiplying a number by its reciprocal gives you 1 (e.g., \(a \times \frac{1}{a} = 1\)). This property is particularly useful when simplifying expressions and solving equations.
In the exercise, when we determined that \((-32)^{3/5} = -8\), taking the reciprocal gives us \((-32)^{-3/5} = \frac{1}{-8} = -\frac{1}{8}\). Understanding reciprocals is vital as it often makes complex expressions more manageable and solvable.