Problem 15 - Algebra and Trigonometry

Question

Evaluate each expression. (a) \(\left(\frac{1}{32}\right)^{2 / 5}\) (b) \((27)^{-4 / 3}\) (c) \(\left(\frac{1}{8}\right)^{-2 / 3}\)

Step-by-Step Solution

Answer

(a) \(\frac{1}{4}\), (b) \(\frac{1}{81}\), (c) \(4\)

Step 1: Expression (a) Simplify the Base

For the expression \(\left(\frac{1}{32}\right)^{2/5}\), first note that \(32 = 2^5\). Consequently, \(\frac{1}{32} = 2^{-5}\). So, the expression simplifies to \((2^{-5})^{2/5}\).

Step 2: Apply the Power of a Power Rule in (a)

Use the rule \((a^m)^n = a^{m \cdot n}\) to simplify \((2^{-5})^{2/5}\) as follows: \((2^{-5})^{2/5} = 2^{-5 \times 2/5} = 2^{-2}\).

Step 3: Evaluate (a) to a Simplified Form

Now calculate \(2^{-2}\): This is \(\frac{1}{2^2} = \frac{1}{4}\). So, the value of \(\left(\frac{1}{32}\right)^{2/5}\) is \(\frac{1}{4}\).

Step 4: Expression (b) Simplify the Base

For \((27)^{-4/3}\), recognize that \(27 = 3^3\). The expression then becomes \((3^3)^{-4/3}\).

Step 5: Apply the Power of a Power Rule in (b)

Using the rule \((a^m)^n = a^{m \cdot n}\), simplify \((3^3)^{-4/3}\) as follows: \((3^3)^{-4/3} = 3^{3 \times (-4/3)} = 3^{-4}\).

Step 6: Evaluate (b) to a Simplified Form

Now, calculate \(3^{-4}\): This gives \(\frac{1}{3^4} = \frac{1}{81}\). So, \((27)^{-4/3} = \frac{1}{81}\).

Step 7: Expression (c) Simplify the Base

For \(\left(\frac{1}{8}\right)^{-2/3}\), notice that \(8 = 2^3\). Thus, \(\frac{1}{8} = 2^{-3}\), and the expression becomes \((2^{-3})^{-2/3}\).

Step 8: Apply the Power of a Power Rule in (c)

Using the rule \((a^m)^n = a^{m \cdot n}\), simplify \((2^{-3})^{-2/3}\) as follows: \((2^{-3})^{-2/3} = 2^{-3 \times (-2/3)} = 2^2\).

Step 9: Evaluate (c) to a Simplified Form

Now, compute \(2^2\): This is equal to \(4\). Therefore, \(\left(\frac{1}{8}\right)^{-2/3}\) is \(4\).

Key Concepts

Power of a Power Rule

When dealing with exponents, the power of a power rule is a powerful tool that can simplify our calculations significantly. This rule states that when you have an exponent raised to another exponent, like \((a^m)^n\), you can multiply these exponents together to obtain the expression \(a^{m \cdot n}\). For example, in the original exercise steps, \((2^{-5})^{2/5}\) was simplified to \(2^{-5 \times 2/5}\) which equals \(2^{-2}\). This rule helps streamline calculations and ensures we avoid long exponent multiplication chains.

The key points to remember with the power of a power rule include:

• Only applicable when both base numbers inside and outside the parenthesis are the same.
• Multiply the exponents directly.
• Results in a single exponent which represents a more simplified expression.

Using this rule correctly can make the process of evaluating complex exponent expressions far less intimidating. Whenever you see a situation involving exponentiation of an exponent, remember this rule to simplify your calculations.

Fractional Exponents

Fractional exponents may seem complex at first glance, but they follow a simple principle that relates them to roots. When you have a fractional exponent such as \(a^{m/n}\), it equates to the \(n\)-th root of \(a\) raised to the power \(m\), that is, \((\sqrt[n]{a})^m\). This implies two steps: first, find the root (denominator of the fraction), then raise it to the power (numerator of the fraction).

Using the previous steps, \((27)^{-4/3}\) was simplified using these principles. Recognizing \(27\) as \(3^3\), it was simplified as \((3^3)^{-4/3}\). Using the power of a power rule, it became \(3^{3 \times (-4/3)} = 3^{-4}\). Finally, \(3^{-4}\) relates back to fractional exponents by treating the negative exponent as indicating a reciprocal and computing as \(\frac{1}{3^4}\).

• Fractional exponents involve roots and powers, understood as two parts.
• Negative fractional exponents denote reciprocals.
• Simplification turns complex expressions into more manageable forms.

Recognizing how fractional exponents stratify an expression can help break down daunting exponent problems into simpler calculations.

Simplifying Expressions

Simplifying expressions, especially ones with complex exponents, involves combining fundamental rules of exponents to a coherent strategy that ensures all expressions are as simple as possible. Each expression should be broken down using exponent rules systematically. In our steps from the original solution:

• Step 1: Simplify the base. Recognize factors that relate bases to smaller powers or simpler numbers, like \(32 = 2^5\).
• Step 2: Apply exponent rules like the power of a power within these smaller, simpler contexts.
• Step 3: Calculate natural values or simplified fractions to determine precise numerical outputs like \(\frac{1}{81}\) from \(3^{-4}\).

Breaking the problem into smaller steps makes it approachable. This involves recognizing the hierarchy and relationships of numbers, applying rules, and executing arithmetic calculations accurately. By simplifying expressions, we pave the way to more manageable problem-solving and clearer mathematical understanding.