Problem 41 - Brief Applied Calculus

Question

Evaluate each expression without using a calculator. $$ (-8)^{-2 / 3} $$

Step-by-Step Solution

Answer

\((-8)^{-2/3} = \frac{1}{4}\).

Step 1: Understanding Negative Exponents

The expression \[(-8)^{-2/3}\] has a negative exponent, which indicates a reciprocal. Rewrite the expression using the reciprocal:\[ (-8)^{-2/3} = rac{1}{(-8)^{2/3}}.\]

Step 2: Applying the Fractional Exponent

The fractional exponent \(2/3\) indicates both a square and a cube root operation. The denominator of 3 represents a cube root, while the numerator of 2 represents a square. Apply the cube root first:\[ \sqrt[3]{-8} = -2.\] Then apply the square:\[ (-2)^2 = 4.\]

Step 3: Combine Both Steps for Final Expression

Combine the reciprocal with the result from the power calculation:\[ \frac{1}{4}.\] Thus, the expression \[ (-8)^{-2/3} = \frac{1}{4}.\]

Key Concepts

Negative Exponents

Negative exponents may seem confusing at first, but they are simpler than you might think. A negative exponent indicates a reciprocal. This means you flip the base and make the exponent positive. For example, \[a^{-b} = \frac{1}{a^b}.\] This is the main rule to remember for negative exponents.

Applying this to the expression \[(-8)^{-2/3},\] we understand that the base, \(-8\), should become its reciprocal, and the exponent becomes positive. Therefore, we rewrite it as \[\frac{1}{(-8)^{2/3}}.\] Then we process the base with its new positive exponent.

Whenever you see a negative exponent:

• Think of the reciprocal.
• Make the exponent positive.
• Solve using the new, positive exponent.

Cube Roots

Cube roots are the inverse operation of cubing a number, and they are used to reduce something to one of its three equal factors. When you see the symbol \(\sqrt[3]{a},\) it asks what number, when multiplied by itself three times, will give \(a.\)

In the case of \(\sqrt[3]{-8},\) we look for a number that satisfies \[x^3 = -8.\] Here, the answer is \(-2\) because \((-2) \times (-2) \times (-2) = -8.\)

When dealing with cube roots, remember:

• They can be negative, meaning they handle negative numbers well.
• Finding a cube root gives you the original factor cubed to get the input value.
• Cubic roots are different from square roots in handling real numbers differently, especially negatives.

Understanding cube roots is essential when dealing with fractional exponents and solving complex problems that involve them.

Reciprocals

The concept of reciprocals is foundational in dividing numbers and understanding exponents. A reciprocal is what you multiply a number by to get \(1.\) Simply put, the reciprocal of any number \(a\) is \(\frac{1}{a}.\)

For positive integers and fractions, this is straightforward. However, applying it to expressions with exponents, like \((-8)^{-2/3},\) requires a small adjustment. The negative in the exponent prompts us to take the reciprocal first, flipping the number, before applying the power calculation.

To effectively work with reciprocals:

• Flip the number to find its reciprocal.
• Apply this concept to simplify expressions with negative exponents.
• Understand that reciprocating is a simple flip operation that allows easier manipulation of expressions.

Grasping how reciprocals work lays the groundwork for simplifying and solving various mathematical problems, especially when dealing with exponents.