Problem 57 - Brief Applied Calculus

Question

Write each expression in power form \(a x^{b}\) for numbers \(a\) and \(b\). $$ \frac{4}{x^{5}} $$

Step-by-Step Solution

Answer

The expression is in power form as \( 4x^{-5} \).

Step 1: Identify the form of the expression

The given expression is \( \frac{4}{x^{5}} \). This is a division of a constant by a variable term raised to a power.

Step 2: Convert to negative exponent form

To express the denominator as a negative exponent in the numerator, use the rule: \( \frac{1}{x^b} = x^{-b} \). Thus, \( \frac{4}{x^5} \) becomes \( 4 \times x^{-5} \).

Step 3: Write in power form

Combine the expression to match the form \( ax^b \). The expression becomes \( 4x^{-5} \), where \( a = 4 \) and \( b = -5 \).

Key Concepts

Understanding Negative Exponents

Negative exponents can be a bit confusing at first, but they follow a simple rule that makes them easy to comprehend. When you see a negative exponent, it essentially means the reciprocal of that base raised to the positive of that exponent.
For instance, when you encounter an expression like \(x^{-b}\), it is equivalent to \(\frac{1}{x^b}\). Therefore, when you see something like \(4 \times x^{-5}\), it's equivalent to putting that \(x^5\) in the denominator, so \(\frac{4}{x^5}\).

• Negative exponents flip the base to the opposite position in a fraction.
• They change the exponent’s sign to positive when moved to the denominator or numerator.

By understanding this rule, you can easily simplify and manipulate expressions with negative exponents.

Exploring Algebraic Expressions

Algebraic expressions allow us to represent mathematical situations using variables, numbers, and operations. They can include constants, like the number 4 in our expression \(4x^{-5}\), and variables, like \(x\). The power form of an algebraic expression is a concise way to write repeated multiplication.
For example, the expression \(4x^{-5}\) means 4 times the reciprocal of \(x\) to the power of 5. In algebra:

• Variables represent numbers whose exact value may be unknown.
• They often come with exponents indicative of their power or reciprocal, depending on whether the exponent is positive or negative.

Algebraic expressions are critical in solving equations, showing relationships, and understanding mathematical models.

The Role of Mathematical Notation

Mathematical notation is a universal language symbolizing mathematical ideas and operations efficiently and consistently. When working with expressions like \(4x^{-5}\), it's important to understand the symbols and convention.
Here’s what to remember:

• Exponents represent the number of times a base is used in multiplication.
• Negative exponents, as noted, symbolize division or reciprocation.
• Parentheses are often used to ensure the correct order of operations.

Understanding mathematical notation helps in deciphering complex expressions and solving them accurately. With consistent practice, you become fluent in this symbolic language, making math simpler.