Problem 21 - Algebra and Trigonometry

Question

\(15-24\) . Evaluate each expression. $$ \begin{array}{llll}{\text { (a) }\left(\frac{1}{32}\right)^{2 / 5}} & {\text { (b) }(27)^{-4 / 3}} & {\text { (c) }\left(\frac{1}{8}\right)^{-2 / 3}}\end{array} $$

Step-by-Step Solution

Answer

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

Step 1: Evaluate part (a)

For part (a), evaluate \( \left(\frac{1}{32}\right)^{2/5} \). Start by expressing \( \frac{1}{32} \) as a power of 2. We note that \( 32 = 2^5 \), so \( \frac{1}{32} = 2^{-5} \). Therefore, \( \left( \frac{1}{32} \right)^{2/5} = (2^{-5})^{2/5} = 2^{-5 \times \frac{2}{5}} = 2^{-2} \). So the result is \( \frac{1}{4} \).

Step 2: Evaluate part (b)

For part (b), evaluate \( (27)^{-4/3} \). First, write 27 as a power of 3: \( 27 = 3^3 \). Therefore, \( (27)^{-4/3} = (3^3)^{-4/3} = 3^{3 \times (-4/3)} = 3^{-4} \). Since \( 3^{-4} = \frac{1}{3^4} = \frac{1}{81} \), the result is \( \frac{1}{81} \).

Step 3: Evaluate part (c)

For part (c), evaluate \( \left(\frac{1}{8}\right)^{-2/3} \). Express \( \frac{1}{8} \) as a power of 2: \( 8 = 2^3 \), so \( \frac{1}{8} = 2^{-3} \). Thus, \( \left( \frac{1}{8} \right)^{-2/3} = (2^{-3})^{-2/3} = 2^{3 \times \frac{2}{3}} = 2^2 = 4 \). So the result is 4.

Key Concepts

Rational Exponents

Rational exponents are an extension of the idea of traditional exponents. They are also called fractional exponents and allow us to express roots and powers in a combined format. For example, an expression such as \( a^{m/n} \) can be understood in two equivalent ways:

• \( (\sqrt[n]{a})^m \): The nth root of \( a \) raised to the mth power.
• \( \sqrt[n]{a^m} \): The nth root of \( a \) raised to the power \( m \).

Applying rational exponents can greatly simplify the process of evaluating roots and powers.

To see this in action, consider part (a) of the exercise: \( \left(\frac{1}{32}\right)^{2/5} \). This expression signifies taking the fifth root of \( \frac{1}{32} \) and then squaring the result. By expressing \( \frac{1}{32} \) as a power of two, it simplifies the calculations using power rules. Understanding rational exponents helps in interpreting and simplifying such expressions.

Negative Exponents

Negative exponents introduce an interesting concept in mathematics. When an exponent is negative, it signifies the reciprocal of the base raised to the absolute value of the exponent.
For instance, \( a^{-n} = \frac{1}{a^n} \). Understanding this helps to translate complex expressions into simpler, more manageable forms.

Part (b) of the exercise highlights this concept. The expression \( (27)^{-4/3} \) involves a negative exponent. Initially, we convert 27 into a power of 3, thereby allowing us to use the properties of exponents. Resulting in \( 3^{-4} \), which means the reciprocal \( \frac{1}{3^4} \), resulting in \( \frac{1}{81} \). Grasping negative exponents allows you to handle inversions of powers effectively.

Power of a Power Rule

When learning about exponents, the power of a power rule often comes in handy. This rule helps simplify expressions like \( (a^m)^n \), by directly multiplying the exponents to get \( a^{m \cdot n} \).
This rule streamlines the process of working with multiple layers of exponents by reducing them to a single exponentiation.

Let's observe part (c) of the exercise: \( \left(\frac{1}{8}\right)^{-2/3} \). Expressing \( \frac{1}{8} \) as \( 2^{-3} \), using the power of a power rule, results in \( (2^{-3})^{-2/3} = 2^{3 \cdot \frac{2}{3}} = 2^2 \). Applying these principles clarifies the expression, revealing that the result is 4. Mastering this rule simplifies exponentiation significantly.