Problem 58 - Applied Calculus

Question

Write each expression in power form \(a x^{b}\) for numbers \(a\) and \(b\). $$ \frac{6}{2 x^{3}} $$

Step-by-Step Solution

Answer

The expression in power form is \(3x^{-3}\).

Step 1: Express the Denominator Using Negative Exponent

The expression \(2x^3\) in the denominator can be represented using a negative exponent in the numerator. This is a basic rule of exponents where \(\frac{1}{x^n} = x^{-n}\). So \(x^3\) in the denominator becomes \(x^{-3}\) when moved to the numerator. This transforms the original expression \(\frac{6}{2x^3}\) to \(\frac{6}{2} \cdot x^{-3}\).

Step 2: Simplify the Coefficient

Next, simplify \(\frac{6}{2}\) by dividing 6 by 2, which results in 3. This means our expression is now \(3x^{-3}\).

Step 3: Identify the Final Power Form

In power form \(ax^b\), the expression \(3x^{-3}\) already fits this structure with \(a = 3\) and \(b = -3\). Thus, the expression in power form is \(3x^{-3}\).

Key Concepts

Understanding Negative Exponents

Negative exponents can seem confusing at first, but they follow a simple rule. When you see a negative exponent, it indicates that the base number or variable is in the denominator of a fraction. For example, if we have the expression \(x^{-n}\), it means \(\frac{1}{x^n}\). It's essentially a way to "move" terms between the numerator and the denominator without changing the value of the expression.
When you encounter a term like \(\frac{1}{x^3}\), by applying the negative exponent rule, it can be rewritten as \(x^{-3}\). This allows for easier manipulation in algebraic expressions and helps simplify complex equations. Remember that flipping a positive exponent into a negative one does not change the term's multiplication or division properties—it's just a convenient way to express it differently.

Simplifying Expressions with Exponents

Simplifying expressions is about making them more manageable and easier to understand. It involves reducing complex terms into simpler forms by applying mathematical rules, such as those dealing with exponents.
With exponents, when you have a term like \(\frac{6}{2x^3}\), you can simplify both the coefficients and the exponents. First, simplify the coefficient \(\frac{6}{2}\), which results in 3. This reduces the complexity of the numerical part of your expression.
Next, deal with the exponents by applying the rule for negative exponents: moving \(x^3\) from the denominator to the numerator changes it to \(x^{-3}\). The whole expression now becomes \(3x^{-3}\), which is its simplified form. This form is often easier to interpret in terms of algebraic operations, such as multiplication and division.

Expressing in Power Form

The power form of an expression is all about neatly organizing it in a standard algebraic format, which makes it easy to understand and use in further calculations. It's written as \(ax^b\), where \(a\) is the coefficient, and \(b\) is the exponent.
Let's break down the final result from our example: \(3x^{-3}\). In this context:

• "3" is the coefficient, representing the number of units we have.
• "\(x\)" is the base, which is the variable in question.
• "-3" is the exponent, indicating that \(x\) is currently placed in the denominator when expressed as \(\frac{1}{x^3}\).

This standardized form simplifies communication in algebraic expressions and equations and makes it easier to perform operations such as multiplication, division, and even advanced calculus tasks. By mastering power form, you set a solid foundation for complex problem-solving.