Problem 39 - Brief Applied Calculus

Question

Evaluate each expression without using a calculator. $$ (-8)^{-1 / 3} $$

Step-by-Step Solution

Answer

The expression evaluates to -1/2.

Step 1: Understanding the Expression

The given expression is (-8)^{-1/3}. This indicates that we must take the cube root of -8 and then find its reciprocal, due to the negative exponent.

Step 2: Finding the Cube Root

First, find the cube root of -8. Since (-2) imes (-2) imes (-2) = -8, the cube root of -8 is -2. Therefore, (-8)^{1/3} = -2.

Step 3: Applying the Negative Exponent

Now, apply the negative exponent rule: (a^{-n} = 1/a^n). Use this rule with the result from the previous step. Therefore, (-8)^{-1/3} = 1/((-8)^{1/3}) = 1/(-2).

Step 4: Final Calculation

Simplify the result: 1/(-2) = -1/2. This is your final result.

Key Concepts

Cube Root

Understanding the cube root is key when dealing with expressions like \((-8)^{-1/3}\). The cube root of a number is a value that, when multiplied by itself twice (a total of three times), gives the original number. In mathematical terms, the cube root of a number \(x\) is written as \(x^{1/3}\). For instance, the cube root of -8 is calculated as follows:

• Identify the number that when multiplied by itself three times results in -8.
• That number is -2, since \((-2) \times (-2) \times (-2) = -8\).

Thus, we find that \((-8)^{1/3} = -2\). This step allows us to simplify expressions and proceed with further calculations accurately.

Reciprocal

The reciprocal of a number is essentially "flipping" the number over. In terms of fractions, the reciprocal of a number \(a\) is \(\frac{1}{a}\).
In our original exercise, once we determined that the cube root of -8 is -2, the next step naturally leads us to its reciprocal because of the negative exponent.

• Due to the negative exponent rule \(a^{-n} = \frac{1}{a^n}\), we take the reciprocal of \(-2\).
• This results in \(\frac{1}{-2}\), also written as \(-\frac{1}{2}\).

Taking reciprocals is crucial to mastering algebraic manipulation, particularly when exponents are involved.

Negative Numbers

Working with negative numbers can sometimes be challenging, but with a bit of practice, it becomes intuitive. In algebra, negative numbers have unique properties:

• When multiplying two negative numbers, the product is positive.
• When dividing or multiplying a negative by a positive, the result remains negative.

In our exercise, dealing with \((-8)\) involves understanding these properties:

• The cube root of -8 is -2 because \((-2)\) multiplied by itself three times results in -8.

Efficient working with negative numbers is essential for solving real-world problems where numbers below zero are part of calculations.

Algebraic Expressions

Algebraic expressions, like \((-8)^{-1/3}\), often combine multiple mathematical concepts. Understanding how these expressions work requires:

• Identifying operations: exponentiation, root extraction, or reciprocal calculation.
• Breaking down the expression into smaller, manageable parts like finding the cube root first.

Once each part is understood, they can be recombined to produce the solution.
In this exercise, handling the negative exponent by sequentially finding the cube root and then applying the reciprocal ensures clarity and accuracy.
This process demonstrates how abstract expressions can be made accessible through systematic breakdown and calculation.