Problem 59 - Algebra for College Students

Question

Simplify each expression. Express final results without using zero or negative integers as exponents. \(\frac{x^{-6}}{x^{-4}}\)

Step-by-Step Solution

Answer

\(\frac{1}{x^2}\)

Step 1: Understand the Problem

We need to simplify the expression \(\frac{x^{-6}}{x^{-4}}\). This involves eliminating negative exponents and expressing the answer in positive exponents without zero exponents.

Step 2: Apply the Quotient Rule for Exponents

The quotient rule for exponents states that \(\frac{a^m}{a^n} = a^{m-n}\). Applying this rule to the expression \(\frac{x^{-6}}{x^{-4}}\), we get \(x^{-6 - (-4)} = x^{-6 + 4}\).

Step 3: Simplify the Exponent

Combine the exponents: \(-6 + 4\) gives us \(-2\). So, the expression simplifies to \(x^{-2}\).

Step 4: Remove the Negative Exponent

To remove the negative exponent, rewrite \(x^{-2}\) as \(\frac{1}{x^2}\), because \(a^{-b} = \frac{1}{a^b}\).

Key Concepts

Negative Exponents

When you see a negative exponent, it can seem a bit tricky at first. But it's quite simple once you get the hang of it! A negative exponent indicates that the base (the number being multiplied) should be on the other side of the fraction line. In other words, if you have something like \(a^{-b}\), it is equivalent to \(\frac{1}{a^b}\). This is because moving the base from one part of a fraction to the other changes the sign of the exponent.

• Negative exponent means: keep the reciprocal of the base.
• Use this to simplify expressions without zero or negative exponents.

This step is crucial when you want to simplify any algebraic expression and present your solution neatly with only positive exponents.

Quotient Rule for Exponents

This rule is a super handy trick that helps when dividing two powers that have the same base. The quotient rule for exponents states that when you divide \(a^m\) by \(a^n\), you can subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\). This is particularly useful when simplifying expressions like \(\frac{x^{-6}}{x^{-4}}\).

• Identify the base, ensuring it is the same for both the numerator and the denominator.
• Subtract the exponent in the denominator from the exponent in the numerator.

For our example, the quotient rule allows us to calculate \(-6 - (-4)\) which results in \(-2\). This part of the process not only simplifies the calculation but also prepares the expression for possible further simplification like converting negative exponents.

Positive Exponents

Once you have simplified the exponent using the rules for exponents, the final step often involves rewriting it in a positive form. That's because standard mathematical convention prefers expressions with positive exponents. A positive exponent simply means the base number is being multiplied by itself For the example of \(x^{-2}\), you can convert it to zero negative exponents by rewriting it as \(\frac{1}{x^2}\).

• A positive exponent equals repeated multiplication.
• Versions with positive exponents are easier to interpret.

By ensuring the final expression has positive exponents, you make it clearer and more universally acceptable. It follows the mathematical norm and helps with further calculations!