Problem 25 - Applied Calculus

Question

Evaluate each expression without using a calculator. $$ (-8)^{5 / 3} $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Expression

The expression we need to evaluate is \((-8)^{5/3}\). This represents a power with a fractional exponent, which can be separated into two parts - a root and a power.

Step 2: Separate the Fractional Exponent

The exponent \(\frac{5}{3}\) can be divided into two operations: raising the number to the power of 5, and taking the cube root. That is, \((-8)^{5/3} = ((-8)^{1/3})^5\).

Step 3: Calculate the Cube Root of -8

Calculate \((-8)^{1/3}\). The cube root of -8 is -2 because \((-2)^3 = -8\). So, \((-8)^{1/3} = -2\).

Step 4: Raise the Result to the Power of 5

Now raise the cube root result to the power of 5: \((-2)^5\). Since \((-2) \times (-2) \times (-2) \times (-2) \times (-2) = -32\), \((-2)^5 = -32\).

Step 5: Conclude the Expression Evaluation

Therefore, the value of \((-8)^{5/3}\) is obtained as follows: first finding the cube root \((-2)\), then raising to the 5th power, resulting in \(-32\).

Key Concepts

Cube Root

Imagine there is a need to find which number, when multiplied by itself three times, gives the original number. This number is called the cube root. For example, the cube root of 8 is 2 because when 2 is multiplied by itself three times, i.e., \(2 \times 2 \times 2 = 8\), it gives us the original number 8. However, when dealing with negative numbers, like -8 in our expression \((-8)^{5/3}\), we apply the same principle. The cube root of -8 is -2, because \((-2) \times (-2) \times (-2) = -8\). Therefore, the cube root can sometimes be a negative number if the original number is negative, maintaining the sign from the original number.

Power and Exponentiation

Understanding powers and exponentiation means understanding how to multiply a number by itself a specified number of times. When we say a number is raised to a certain power, like the power of 2, it means multiplying that number by itself. For instance, \(3^2\) is equal to \(3 \times 3 = 9\).

In the case of fractional exponents, such as \(5/3\), exponentiation involves two steps:

• Finding the root of the number (in this case, a cube root).
• Raising the result to the remaining power part (in this case, raising to the 5th power).

Fractional exponents break the process into these two parts. With \((-8)^{5/3}\), once we compute the cube root, which is -2, the next step is raising \(-2\) to the power of 5, which gives us the result \(-32\).

Negative Base Powers

Working with negative bases introduces additional considerations, particularly with odd and even exponents. When a negative number is raised to an odd power, the result remains negative because the product of an odd number of negative factors is negative.

For example, \((-2)^5\) results in

• \((-2) \times (-2) = 4\)
• \(4 \times (-2) = -8\)
• \(-8 \times (-2) = 16\)
• \(16 \times (-2) = -32\)

Here, because we have an odd number of negative multiplications (five, in this case), the result is \(-32\). However, if the exponent was even, like 2 or 4, the result would be positive because pairs of negatives multiply together to become positive. This understanding is crucial when dealing with expressions like \((-8)^{5/3}\) where the base is negative and the power involves multiplying several negative factors.