Problem 95 - Elementary Algebra

Question

\(\frac{x^{9}}{x^{5}}\)

Step-by-Step Solution

Answer

The simplified expression is \(x^{4}\).

Step 1: Understand the Problem

The problem asks to simplify the expression \(\frac{x^{9}}{x^{5}}\). This involves dividing two exponential terms with the same base.

Step 2: Apply the Exponent Rule

When dividing exponents with the same base, subtract the exponent in the denominator from the exponent in the numerator. The general rule is \(\frac{a^{m}}{a^{n}} = a^{m-n}\).

Step 3: Simplify the Expression

Using the exponent rule, subtract the exponent in the denominator from the exponent in the numerator: \(\frac{x^{9}}{x^{5}} = x^{9-5} = x^{4}\).

Key Concepts

Understanding Exponent Rules

When working with exponents, it’s essential to grasp the basic rules governing them. An exponent tells you how many times to multiply a base number by itself. For example, in the expression \(x^3\), the base is \(x\) and the exponent is 3, meaning \(x\) is multiplied by itself three times: \(x \times x \times x\).
One of the fundamental exponent rules is that when you raise a power to another power, you multiply the exponents: \((a^m)^n = a^{m \times n}\).
Another key rule is the product of powers rule, where you add the exponents when multiplying like bases: \(a^m \times a^n = a^{m+n}\). Keep these rules in mind, as they are the building blocks for simplifying exponential expressions.

Division of Exponents

Now, let's discuss the rule for dividing exponents, which is particularly relevant to our exercise. When dividing exponential expressions with the same base, you subtract the exponent in the denominator from the exponent in the numerator.
The mathematical rule to remember is: \(\frac{a^m}{a^n} = a^{m-n}\).
In simple terms, if you are dividing \(x^m\) by \(x^n\), you subtract \(n\) from \(m\) and keep the base \(x\).
This rule works because you are effectively canceling out common factors in the numerator and the denominator. For example, in the expression \(\frac{x^9}{x^5}\), you subtract 5 from 9 to get \(x^{9-5} = x^4\).

Simplification Made Easy

Simplification is the process of making an expression less complicated. When simplifying expressions with exponents, always look for opportunities to apply the exponent rules.
In the exercise \(\frac{x^9}{x^5}\), simplification involves recognizing that both the numerator and the denominator have the same base, which is \(x\).
Using the division of exponents rule, subtract the exponent in the denominator from the exponent in the numerator: \(\frac{x^9}{x^5} = x^{9-5} = x^4\).
The simplified form is \(x^4\), which is much easier to understand and use in further calculations.
Simplification not only makes expressions cleaner, but it also makes them more workable, especially in complex algebraic problems. Always remember to apply the core exponent rules to make your work simpler and clearer.