Problem 31 - Applied Calculus

Question

Evaluate each expression without using a calculator. $$ \left(\frac{1}{32}\right)^{2 / 5} $$

Step-by-Step Solution

Answer

The value of \(\left(\frac{1}{32}\right)^{2 / 5}\) is \(\frac{1}{4}\).

Step 1: Rewrite the Base as a Power of a Prime

The number 32 can be rewritten as a power of 2. Since \(32 = 2^5\), we have \(\frac{1}{32} = 2^{-5}\). Thus, we can express the given expression as \(\left(2^{-5}\right)^{2/5}\).

Step 2: Apply the Power of a Power Rule

Use the exponentiation rule \((a^m)^n = a^{m\cdot n}\). For the expression \(\left(2^{-5}\right)^{2/5}\), apply this rule to get:\[2^{-5\cdot \frac{2}{5}}\].

Step 3: Simplify the Exponents

Calculate the product in the exponent: \(-5 \times \frac{2}{5} = -2\). So, the expression becomes \(2^{-2}\).

Step 4: Evaluate the Simplified Expression

To find \(2^{-2}\), take the reciprocal of \(2^2\). Since \(2^2 = 4\), the expression evaluates to \(\frac{1}{4}\).

Step 5: Verify the Final Result

Verify the simplification by confirming each step, ensuring that the power, power rule, and simplification were correctly done from steps 1 to 4. Our final result matches as \(\frac{1}{4}\).

Key Concepts

Power of a Power Rule

The Power of a Power Rule is a fundamental concept in exponentiation. It states that when you raise a power to another power, you multiply the exponents. This can be expressed mathematically as \((a^m)^n = a^{m \cdot n}\). Let's break down what this means:

• "\(a\)" is the base of the expression.
• "\(m\)" is the exponent of the base.
• "\(n\)" is the exponent applied to the initial power \(a^m\).

When you apply the Power of a Power Rule, you combine both exponents into a single exponentiation. For example, in the expression \((2^{-5})^{2/5}\), you multiply the exponents \(-5\) and \(2/5\) together to get \(-2\). The rule simplifies complex exponentiation tasks by converting multi-layered powers into a straightforward multiplication. This method helps manage and simplify expressions easily, making even intimidating equations more approachable.

Negative Exponents

Negative exponents may seem confusing at first glance, but they are simply another way to express the reciprocal of a number. A negative exponent indicates that you take the reciprocal of the base raised to the absolute value of the exponent. In mathematical terms, \(a^{-m} = \frac{1}{a^m}\). Here's how it works:

• "\(a\)" is your base number.
• "\(-m\)" is the negative exponent.

Instead of thinking of a negative exponent as indicating a negative number, consider it as directing you to find the reciprocal of the positive exponent. For instance, \(2^{-2}\) means you take the reciprocal of \(2^2\). Calculating \(2^2\) gives 4, and taking the reciprocal provides \(\frac{1}{4}\).
Understanding negative exponents helps make sense of expressions involving small fractions, as it provides a systematic approach for conversion into manageable forms.

Fractional Exponents

Fractional exponents, though possibly intimidating, are quite straightforward and often relate to roots. A fractional exponent represents both an exponentiation and a root, and is expressed as \(a^{m/n}\). This indicates that you perform two actions: raise \(a\) to the \(m\)-th power, and then take the \(n\)-th root. Alternatively, you can take the root first and then raise the result to the power.
Here's a breakdown of \(a^{m/n}\):

• "\(a\)" is the base of the expression.
• "\(m\)" is the power to which the base is raised.
• "\(n\)" is the root taken of the result.

For example, in the expression \((\frac{1}{32})^{2/5}\), the 2 acts as a power, and the 5 suggests a 5th root. First, you can transform \(\frac{1}{32}\) into \(2^{-5}\), and then simplify \((2^{-5})^{2/5}\) using the Power of a Power Rule. The fractional exponent \(2/5\) efficiently organizes these operations, indicating the sequence and relationships between powers and roots. This simplifies expressions distinctly, merging the complexity of roots and powers into a clear, instructive form.