Problem 41 - Applied Calculus

Question

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

Step-by-Step Solution

Answer

The value of \((-8)^{-2/3}\) is \(\frac{1}{4}\).

Step 1: Rewrite the negative exponent

The expression \[ (-8)^{-2/3} \]can be rewritten using the property of negative exponents, which states that any negative exponent can be converted to a positive exponent by taking the reciprocal. So, \[ (-8)^{-2/3} = \frac{1}{(-8)^{2/3}}. \]

Step 2: Understand the meaning of fraction exponents

The fraction exponent \(-2/3\) indicates two operations: the denominator \(3\) suggests we take the cube root, and the numerator \(2\) indicates we then square the result. Therefore, we need to find \[ (-8)^{2/3} = \left((-8)^{1/3}\right)^2. \]

Step 3: Find the cube root

The first step in calculating \((-8)^{2/3}\) is to find the cube root of \(-8\). Since \(-2)^3 = -8\), we find that\[ (-8)^{1/3} = -2. \]

Step 4: Square the result

Next, we square \(-2\) to complete the expression \((-8)^{2/3}\):\[ (-2)^2 = 4. \]

Step 5: Write the reciprocal

Now, substitute back into the reciprocal expression, \((-8)^{-2/3}\) becomes:\[ \frac{1}{(-8)^{2/3}} = \frac{1}{4}. \]

Key Concepts

Negative Exponents

When dealing with negative exponents, it helps to remember a general rule:

• A negative exponent indicates that we take the reciprocal of the base raised to the corresponding positive exponent.
• For example, \[ a^{-n} = \frac{1}{a^{n}} \quad \text{for any non-zero } a. \]

In our exercise, we apply this rule to the term \((-8)^{-2/3}\) to transform it into \(\frac{1}{(-8)^{2/3}}.\) This simplifies our calculations by turning the original expression into a fraction with a positive exponent. Remember, handling fractions often makes complex operations more manageable!
Keeping this rule in mind will serve you well across different mathematical contexts, where converting to positive exponents simplifies the process.

Cube Root

The cube root is a special kind of root related to the power of three. It determines a number that, when multiplied by itself twice (three times in total), results in the original number.

• Mathematically, for a number \( x, \)\( x^{1/3} \)represents the cube root.
• The cube root of \(-8 \)is found by determining which number cubed gives us \(-8. \)

In this exercise, the cube root of \(-8\)is \(-2,\)since \((-2) \cdot (-2) \cdot (-2) = -8.\)
The practice of finding cube roots is an essential skill in algebra, especially when dealing with fraction powers and simplifying expressions involving roots.

Exponentiation Rules

Exponentiation rules are the backbone of simplifying and solving expressions involving powers. Here are some vital rules:

• Product of Powers: When multiplying like bases, you add exponents: \( a^{m} \cdot a^{n} = a^{m+n}. \)
• Power of a Power: When raising a power to another power, you multiply exponents: \( (a^{m})^{n} = a^{m \cdot n}. \)
• Power of a Product: The exponent applies to each factor within the parentheses: \( (ab)^{n} = a^{n} \cdot b^{n}. \)

In our original problem, this concept guided steps, especially where we worked with fractional exponents.We leveraged these rules to break down \((-8)^{2/3}\)into more manageable parts: first finding the cube root and then squaring the result.
Understanding these rules allows you to navigate through complex exponentiation scenarios with ease and confidence.