Problem 94 - Algebra for College Students

Question

How would you convince someone that \(x^{6} \div x^{2}\) is \(x^{4}\) and not \(x^{3} ?\)

Step-by-Step Solution

Answer

By subtracting exponents: \(6 - 2 = 4\), hence \(x^{4}\).

Step 1: Understand the Problem

We are given the expression \(x^{6} \div x^{2}\) and need to explain how it simplifies to \(x^{4}\). The confusion might arise from mistaking exponent division rules.

Step 2: Apply the Division Rule for Exponents

When dividing like bases with exponents, subtract the exponent in the denominator from the exponent in the numerator: \(x^{a} \div x^{b} = x^{a-b}\).

Step 3: Execute the Simplification

Apply the exponent division rule to \(x^{6} \div x^{2}\): subtract the exponent in the denominator (2) from the exponent in the numerator (6): \(x^{6-2} = x^{4}\).

Step 4: Confirm the Result

Verify that the subtraction of exponents was performed correctly: \(6-2 = 4\), confirming \(x^{4}\) is correct, not \(x^{3}\).

Key Concepts

Division of Exponents

One of the most important rules in math when dealing with exponents is the division of exponents. When you have an expression where the same base is being divided, like in the example \( x^6 \div x^2 \), you can simplify by subtracting the exponents. Here's the simple rule:

• For the expression \( x^a \div x^b \), take the exponent from the denominator \( b \) and subtract it from the exponent in the numerator \( a \). So, \( x^a \div x^b = x^{a-b} \).
• This means you only subtract the numbers above and below the same base.

This rule is used because division and multiplication are inverse operations, and subtracting the exponents makes sense because you are essentially canceling out repeated multiplication. This simple subtraction gives you the new exponent. Hence, \( x^6 \div x^2 \) results in \( x^{6-2} \), which simplifies to \( x^4 \). This is a quick and efficient method to simplify expressions with exponents.

Exponent Rules

Exponent rules are guidelines that help us manipulate expressions with exponents easily. They are key in algebra and can simplify seemingly complex problems.

When working with exponents, remember:

• Product of Powers Rule: \( x^a \times x^b = x^{a+b} \). You add exponents when multiplying like bases.
• Power of a Power Rule: \( (x^a)^b = x^{a\times b} \). You multiply exponents when a power is raised to another power.
• Division of Powers Rule: As discussed, \( x^a \div x^b = x^{a-b} \). Subtract exponents to divide.
• Zero Exponent Rule: \( x^0 = 1 \) for any non-zero base \( x \), meaning any number raised to the power of zero is 1.

These rules are universally applicable in algebra. They allow for the manipulation of terms in expressions, making it much easier to solve equations and simplify results. By mastering these, calculations involving exponents become straightforward and manageable.

Algebra Simplification

Algebra simplification is a process used to reduce an expression down to its simplest form. This involves using various algebraic rules, like exponent rules, and basic arithmetic operations. Simplification helps to understand expressions more clearly.

To simplify an expression:

• Apply basic arithmetic operations such as addition, subtraction, multiplication, and division where possible.
• Use algebraic rules, such as exponent rules, to reduce terms in the expression.
• Combine like terms where applicable to further reduce the expression.

For example, with \( x^6 \div x^2 \), we apply the division of exponents rule to simplify it to \( x^4 \). Be sure to perform each step correctly and verify your work as done in the original exercise solution. The right approach ensures that you simplify correctly and arrive at the correct answer with confidence. Whether you're working on simple or complex expressions, the goal is to make them as clear and straightforward as possible.