Problem 76 - Precalculus: Functions and Graphs

Question

Express as a quotient. $$x^{-23}+x^{70}$$

Step-by-Step Solution

Answer

\(x^{-23} + x^{70} = \frac{1 + x^{93}}{x^{23}}\).

Step 1: Identify the Expression

We need to express the given expression \(x^{-23} + x^{70}\) as a quotient. Observe that we have terms with negative and positive exponents.

Step 2: Convert Negative Exponent

Recall that a negative exponent \(a^{-b}\) can be rewritten as \(\frac{1}{a^b}\). Rewrite \(x^{-23}\) using this property: \(x^{-23} = \frac{1}{x^{23}}\).

Step 3: Common Denominator

To express as a single quotient, we need a common denominator for the terms \(\frac{1}{x^{23}}\) and \(x^{70}\). Note that \(x^{70}\) can be expressed as \(\frac{x^{70}}{1}\). The least common denominator would be \(x^{23}\), reshaping \(x^{70} = x^{70}\cdot \frac{x^{23}}{x^{23}} = \frac{x^{93}}{x^{23}}\).

Step 4: Express as Quotient

With a common denominator, add the terms: \(\frac{1}{x^{23}} + \frac{x^{93}}{x^{23}} = \frac{1 + x^{93}}{x^{23}}\). Now, the expression \(x^{-23} + x^{70}\) is successfully expressed as the quotient \(\frac{1 + x^{93}}{x^{23}}\).

Key Concepts

Negative Exponents

Negative exponents often initially confuse students, but they are rather simple once you grasp the core concept. When we see an exponent with a negative sign, like in the expression \(x^{-23}\), it signals that we need to take the reciprocal of the base raised to the positive of that exponent.

• A negative exponent \(a^{-b}\) becomes \(\frac{1}{a^b}\).
• This means for our example, \(x^{-23} = \frac{1}{x^{23}}\).

By converting negative exponents into fractions, complex expressions become much easier to work with. This transformation helps to simplify expressions so that they can be added, subtracted, or otherwise manipulated easily.

Quotient Form

Expressing an algebraic expression as a quotient involves rewriting it as a fraction. In our example, we started with \(x^{-23} + x^{70}\) and were tasked with rewriting it in quotient form.

• Initially, we rewrite the negative exponent as \(\frac{1}{x^{23}}\).
• The second term, \(x^{70}\), is rewritten as a fraction by expressing it over 1: \(\frac{x^{70}}{1}\).

The goal when rewriting these expressions is to make sure each term has a common structure—here, fractions—enabling us to combine them under a single denominator.

Common Denominator

One of the main hurdles in combining fractions is finding a common denominator. It allows us to add or subtract fractions smoothly. When dealing with algebraic expressions, this principle remains unchanged. In our exercise, we want to add \(\frac{1}{x^{23}}\) and \(\frac{x^{70}}{1}\), requiring a least common denominator. Here are the steps we followed:

• The common denominator chosen was \(x^{23}\).
• This meant converting \(\frac{x^{70}}{1}\) into \(\frac{x^{93}}{x^{23}}\), a process involving multiplying the numerator and denominator by \(x^{23}\).

With common denominators, our fractions now share the same base in the denominator, allowing for easy combination into \(\frac{1 + x^{93}}{x^{23}}\).

Exponent Rules

Understanding the rules governing exponents is key to manipulating and simplifying expressions accurately. In this problem, several rules were used to achieve the goal.

• **Product of Powers Rule:** This rule states that \(a^m \cdot a^n = a^{m+n}\). We used this when factoring powers in the denominator to add powers together.
• **Power of a Power Rule:** Expresses that \((a^m)^n = a^{mn}\), useful for managing expressions with nested exponents though not directly used in this exercise.

Exponent rules are like the bedrock for algebraic simplification, aiding us in converting negatives to positive exponents, bringing expressions into a common form, and simplifying terms. Knowing how to apply these rules is crucial for mastery of algebraic expressions.