Problem 65 - Algebra 1: Concepts and Skills

Question

Simplify the expression \(\frac{x^{-9}}{x^{-3}}\) A. \(\frac{1}{x^{-6}}\) B. \(\frac{1}{x^{6}}\) C. \(x^{-3}\) D. \(x^{3}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{x^{6}}\)

Step 1: Setup

Our starting expression is \(\frac{x^{-9}}{x^{-3}}\)

Step 2: Apply the rule of exponents for division

When dividing numbers which both have exponents, subtract the denominator's exponent from the numerator's exponent. Thus, we have \(x^{-9-(-3)}\).

Step 3: Simplify the exponent

Subtracting a negative is the same as adding so the equation becomes \(x^{-9+3}\) which simplifies further to \(x^{-6}\).

Step 4: Convert the negative exponent to a positive exponent

The rule for a negative exponent is to take the reciprocal to make the exponent positive. Therefore, \(x^{-6}\) becomes \(\frac{1}{x^{6}}\).

Key Concepts

Understanding Negative Exponents

Negative exponents can seem a bit confusing at first, but they follow a straightforward rule. When you have a negative exponent, it means you're taking the reciprocal of the base raised to the positive of that exponent. In simpler terms,

• A negative exponent signifies division; it's the inverse of multiplying by that base. For example, with something like \(x^{-3}\), it would be rewritten as \(\frac{1}{x^3}\).
• Think of negative exponents as 'flipping' the base to the other side of a fraction, turning the exponent positive in the process.

Whenever you encounter a negative exponent, remember that you can convert it to a positive exponent by using the reciprocal. This makes calculations much simpler and less error-prone.

Simplifying Expressions with Exponents

Simplifying expressions with exponents involves applying specific exponent rules to reduce the complexity of the expression. To simplify properly, you need to:

• Use multiplication or division rules for exponents. In multiplication, add the exponents if the bases are the same. In division, which is especially important for this exercise, you subtract the exponents.
• Look for patterns or structures within the expression that can make simplification easier. For instance, \(x^{-9}/x^{-3}\) involves subtraction of exponents: \(x^{-9 - (-3)}\) becomes \(x^{-6}\).
• Apply any additional rules necessary, such as converting negative exponents to positive ones by finding reciprocals.

The goal is to rewrite the expression in its simplest form, making it clearer and often more useful for further operations.

Mastering Division of Exponents

Division of exponents is a common operation in algebra that often trips students up. Fortunately, there's a simple rule to follow. When dividing expressions with the same base, subtract the exponent of the denominator from the exponent of the numerator. For our case:

• With \(\frac{x^{-9}}{x^{-3}}\), we subtract the lower exponent \((-3)\) from the upper exponent \((-9)\).
• This subtraction simplifies to \(x^{-9 + 3}\), which evaluates to \(x^{-6}\).
• The final step would be turning the result to a positive exponent, often through reciprocals, giving us \(\frac{1}{x^{6}}\).

Always focus on keeping track of your signs—whether you’re subtracting a negative or positive—to avoid mistakes.