Problem 43 - Algebra for College Students

Question

For Problems \(43-62\), simplify each expression. Express final results without using zero or negative integers as exponents. \(x^{2} \cdot x^{-8}\)

Step-by-Step Solution

Answer

\(\frac{1}{x^6}\)

Step 1: Recall the Rule of Exponents for Multiplication

When multiplying powers with the same base, we add the exponents. This is an essential rule of exponents to remember: If you have \(a^m \cdot a^n\), then the result is \(a^{m+n}\).

Step 2: Apply the Exponent Rule

Given the expression \(x^2 \cdot x^{-8}\), apply the multiplication of exponents rule: \(x^{2 + (-8)} = x^{-6}\).

Step 3: Convert Negative Exponents to Positive

To express the result without negative exponents, recall that \(x^{-n} = \frac{1}{x^n}\). Thus, \(x^{-6}\) becomes \(\frac{1}{x^6}\).

Key Concepts

Rule of Exponents

Understanding the rule of exponents is essential when simplifying expressions that involve powers. This rule states that when you multiply numbers with the same base, you simply add their exponents together. For instance, if you have the expression \(a^m \cdot a^n\), you can simplify this to \(a^{m+n}\). This means you increase the power of the base by summing the individual exponents.

• Remember: The base must be the same for the rule to apply.
• This rule helps in reducing complex expressions into simpler forms quickly.

In our exercise, the expression \(x^2 \cdot x^{-8}\) initially looks complicated, but by applying this rule, you sum the exponents: \(2 + (-8)\), resulting in \(x^{-6}\). This is a straightforward process once you grasp the rule of combining exponents by addition.

Positive Exponents

Converting expressions to have only positive exponents is often necessary for simplification and clarity. A positive exponent signifies how many times a base is multiplied by itself, whereas a negative exponent indicates division or reciprocal multiplication.

• A negative exponent, like \(x^{-n}\), translates to \(\frac{1}{x^n}\), converting multiplication into division.
• Only positive exponents are used in final answers for ease of interpretation and to meet standard mathematical practices.

In our example, the expression \(x^{-6}\) was converted using this principle to \(\frac{1}{x^6}\). This step ensures that the presentation of the result adheres to the convention of using only positive exponents.

Multiplying Powers

When multiplying powers, especially those with the same base, understanding how exponents operate is crucial. The multiplication of powers is simplified by using the addition rule of exponents, as we discussed earlier.

• Multiplying means adjusting the exponent, but the base remains the same throughout the calculation.
• This helps manage expressions that initially seem unwieldy, breaking them down into manageable components.

For the problem \(x^2 \cdot x^{-8}\), we recognize that both terms share the same base of \(x\). Thus, by adding the exponents 2 and \(-8\), we derived \(x^{-6}\). Simplifying using this method allows you to easily handle more complex algebraic expressions involving multiple powers.