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Chapter 5: Problem 15

\(\frac{x^{9}}{x^{2}}\)

Short Answer

Expert verified
The solution is \(x^7\).

Step by step solution

01

Apply the Quotient Rule of Exponents

When dividing exponential terms with the same base, subtract the exponent of the denominator from the exponent of the numerator. The general rule is: \[\frac{a^m}{a^n} = a^{m-n}\]
02

Substitute the Given Exponents

Using the given expression \[\frac{x^9}{x^2}\], apply the quotient rule: \[9 - 2\]
03

Simplify the Exponent

Perform the subtraction to simplify the exponent: \[9 - 2 = 7\]
04

Write the Final Expression

Combine the base with the simplified exponent: \[x^7\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Division
Exponential division involves dividing expressions that have the same base but different exponents. In mathematics, the concept is simplified by using the Quotient Rule of Exponents. This rule allows you to subtract the exponent in the denominator from the exponent in the numerator. For example, in the expression \(\frac{x^9}{x^2}\), you would subtract the exponent 2 from the exponent 9. The rule can be summarized as \(\frac{a^m}{a^n} = a^{m-n}\). It makes handling divisions of exponentials easier and less prone to errors.
Simplifying Exponents
Simplifying exponents means reducing them to their simplest form. When working with the expression \(\frac{x^9}{x^2}\), apply the Quotient Rule of Exponents to simplify it. Start by subtracting the exponent in the denominator from the exponent in the numerator: \9 - 2\. This will give you the simplified exponent \7\. Therefore, the original expression \(\frac{x^9}{x^2}\) can be rewritten as \x^7\. Simplifying exponents helps to make mathematical expressions more manageable and easier to work with in further calculations.
Basic Algebra
Basic algebra includes fundamental rules and concepts that are the building blocks for more advanced mathematics. One such concept is the manipulation of exponential terms. Here, we focused on an important algebraic rule known as the Quotient Rule of Exponents, which is a powerful tool in simplifying algebraic fractions. Using this rule, you convert the division of like bases with exponents into simpler forms by subtracting exponents. This simplification process helps to keep equations easier to manage. Mastering it is crucial for solving more complex algebraic problems in the future.

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