Problem 14 - Algebra and Trigonometry

Question

Evaluate each expression. (a) \(1024^{-0.1}\) (b) \(\left(-\frac{27}{8}\right)^{2 / 3}\) (c) \(\left(\frac{25}{64}\right)^{3 / 2}\)

Step-by-Step Solution

Answer

(a) \( \frac{1}{2} \); (b) \( \frac{9}{4} \); (c) \( \frac{125}{512} \).

Step 1: Simplifying Expression (a)

To evaluate the expression \(1024^{-0.1}\), we start by rewriting the negative exponent. A negative exponent indicates the reciprocal, so \(x^{-a} = \frac{1}{x^a}\). Therefore, \(1024^{-0.1} = \frac{1}{1024^{0.1}}\).

Step 2: Evaluate Exponent for Expression (a)

Now, calculate \(1024^{0.1}\). Recognize that the base 1024 is a power of 2: \(1024 = 2^{10}\). Hence, \(1024^{0.1} = (2^{10})^{0.1} = 2^{10 \times 0.1} = 2^1 = 2\). So, \(1024^{-0.1} = \frac{1}{2}\).

Step 3: Simplifying Expression (b)

For \(\left(-\frac{27}{8}\right)^{2/3}\), recognize that \(x^{m/n} = (x^m)^{1/n} = \sqrt[n]{x^m}\). Thus, \(\left(-\frac{27}{8}\right)^{2/3} = \sqrt[3]{\left(-\frac{27}{8}\right)^2}\).

Step 4: Evaluate Exponent for Expression (b)

First, calculate \(\left(-\frac{27}{8}\right)^2\). Squaring leads to \(\left(\frac{27}{8}\right)^2 = \frac{729}{64}\). Next, compute the cube root: \(\sqrt[3]{\frac{729}{64}} = \frac{\sqrt[3]{729}}{\sqrt[3]{64}} = \frac{9}{4}\).

Step 5: Simplifying Expression (c)

For \(\left(\frac{25}{64}\right)^{3/2}\), use \(x^{m/n} = (x^m)^{1/n} = \sqrt[n]{x^m}\). Thus, \(\left(\frac{25}{64}\right)^{3/2} = \left(\sqrt{\frac{25}{64}}\right)^3\).

Step 6: Evaluate Exponent and Simplify for Expression (c)

Find the square root: \(\sqrt{\frac{25}{64}} = \frac{\sqrt{25}}{\sqrt{64}} = \frac{5}{8}\). Then, raise to the power of 3: \(\left(\frac{5}{8}\right)^3 = \frac{125}{512}\).

Key Concepts

Rational Exponents

Rational exponents, also known as fractional exponents, represent powers in the form of fractions. They are expressed as \( x^{m/n} \), which denotes \( n \)-th root of \( x \) raised to the \( m \) power. Here’s how we can break it down:

• The denominator \( n \) indicates the root to be taken, such as a square root \( (n = 2) \), cube root \( (n = 3) \), etc.
• The numerator \( m \) specifies the power to which the base \( x \) is raised.

Understanding rational exponents is essential to simplifying expressions that involve roots and powers, as demonstrated when solving parts (b) and (c) of the exercise. In these exercises, \( \left(-\frac{27}{8}\right)^{2/3} \) signifies taking the cube root of the square of \( -\frac{27}{8} \), while \( \left(\frac{25}{64}\right)^{3/2} \) means raising the square root of \( \frac{25}{64} \) to the power of 3.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the corresponding positive exponent. When you see an expression like \( x^{-a} \), it means \( \frac{1}{x^a} \). This is crucial for transforming expressions with negative powers into a more workable form.
In the original exercise, \( 1024^{-0.1} \) is simplified initially to \( \frac{1}{1024^{0.1}} \) based on this concept. Converting negative exponents helps in understanding and simplifying complex expressions, especially when the next steps involve simpler calculations like root extraction or power raising.

Reciprocal

The reciprocal of a number is simply \( \frac{1}{x} \), where \( x \) is not zero. This mathematical concept is deeply intertwined with negative exponents. When a number or expression is raised to a negative exponent, finding its reciprocal is the first step in simplifying it.
For instance, when we encounter \( 1024^{-0.1} \) in the exercise, taking the reciprocal is the initial step that helps change and simplify the expression into \( \frac{1}{1024^{0.1}} \). This transformation is crucial for handling equations and expressions, especially in algebra and calculus, where negative exponents frequently occur.

Cube Root

The cube root of a number is what you multiply by itself three times to get that number. It is represented using the symbol \( \sqrt[3]{x} \) or equivalently, as \( x^{1/3} \). Calculating a cube root is essential when dealing with fractional exponents of the form \( x^{m/3} \).
In the exercise given, calculating the cube root is fundamental to solving expression (b), \( \left(-\frac{27}{8}\right)^{2/3} \), after the squaring step: \( \sqrt[3]{\frac{729}{64}} \). This operation simplifies down to \( \frac{9}{4} \), demonstrating the utility and necessity of understanding cube roots in exponent-related problems.