Problem 5 - Algebra 2 and Trigonometry

Question

Simplify each expression. In each exercise, all variables are positive. \(x^{6} \div x^{2}\)

Step-by-Step Solution

Answer

The expression simplifies to \(x^4\).

Step 1: Identify the Given Expression

The expression given is \(x^{6} \div x^{2}\). This means we have \(x^6\) being divided by \(x^2\).

Step 2: Apply the Quotient Rule for Exponents

Recall that the quotient rule for exponents states \(\frac{a^m}{a^n} = a^{m-n}\) for the same base \(a\). In this case, the base is \(x\), so apply the rule: \(x^{6-2}\).

Step 3: Perform the Subtraction

Subtract the exponents: \(6 - 2 = 4\), which gives \(x^4\).

Step 4: Write the Simplified Expression

After applying the quotient rule, the expression simplifies to \(x^4\).

Key Concepts

Quotient Rule for Exponents

In algebra, simplifying expressions with exponents often involves using rules that help us handle operations like multiplication and division. One such rule is the **Quotient Rule for Exponents**. This rule is vital when working with fractions of exponential expressions.
The quotient rule states: \[\frac{a^m}{a^n} = a^{m-n}\] where \(a\) is the common base, and \(m\) and \(n\) are the exponents. It tells us that when dividing like bases, you can subtract the exponent in the denominator from the exponent in the numerator.

Here's why this works:

• Both \(a^m\) and \(a^n\) represent repeated multiplication of the base \(a\).
• In a fraction, dividing by \(a^n\) cancels out \(n\) factors of \(a\) from the numerator.
• What remains is the base \(a\) raised to the power of \(m-n\).

This is a quick and efficient way to simplify expressions involving division when the bases are the same.

Positive Variables

When simplifying algebraic expressions, the assumption that all variables are positive plays a critical role. This assumption simplifies the calculations and avoids issues related to negative bases.

With positive variables:

• You avoid sign changes that commonly arise with negative numbers.
• Calculations become straightforward because negative exponents do not have to be considered conversely.
• Expressions like \(x^{-n}\) or \(y^{-3}\) aren't problematic because positive variables ensure the base does not invert or require more complex transformations.

For example, in an expression like \(x^6 \div x^2\), just knowing \(x\) is positive allows us to easily carry out simplification using the quotient rule without worrying about negative or zero results from basic operations.

Exponent Subtraction

Exponent subtraction is an essential concept in simplifying expressions through division.Once the quotient rule for exponents is applied, subtracting the exponents is the next critical step.

Here's how it works:

• Identify the exponents of the base in both the numerator and the denominator.
• Subtract the exponent in the denominator from the exponent in the numerator.
• The result becomes the new exponent for that base in the simplified expression.

In our example \(x^6 \div x^2\), exponent subtraction is applied after applying the quotient rule:

• Subtract 2 from 6.
• This gives us \(4\), so the expression simplifies to \(x^4\).

This subtraction is straightforward and provides a direct path to the simplified expression, making calculations quicker and easier.