Problem 60 - Algebra for College Students

Question

Simplify each expression. Express final results without using zero or negative integers as exponents. \(\frac{a^{-2}}{a^{2}}\)

Step-by-Step Solution

Answer

\(\frac{1}{a^4}\)

Step 1: Apply the Quotient of Powers Rule

The expression \(\frac{a^{-2}}{a^{2}}\) can be simplified using the quotient of powers rule, which states \(\frac{b^m}{b^n} = b^{m-n}\). Here, \(m = -2\) and \(n = 2\). So, apply the rule: \[ a^{-2 - 2} = a^{-4} \]

Step 2: Convert Negative Exponent to Positive

To express the final result with non-negative exponents, convert the negative exponent to a positive by taking the reciprocal. For \(a^{-4}\), this means:\[ a^{-4} = \frac{1}{a^4} \]

Key Concepts

Negative Exponents

When you see a negative exponent in mathematics, it may look confusing at first, but it simply means you have to perform a small transformation. A negative exponent indicates that you should take the reciprocal of the base and then apply the positive exponent. For example, take the expression \( a^{-4} \). Here, the base is \( a \) and the exponent is \( -4 \). The negative sign tells you to find the reciprocal of \( a \) raised to the positive 4.

• \( a^{-n} = \frac{1}{a^{n}} \)
• Turn the exponent positive by swapping the position of the base (or moving it to the other side of the fraction).

In our case, \( a^{-4} \) becomes \( \frac{1}{a^4} \) after finding the reciprocal. It's like flipping the base from the numerator to the denominator or vice-versa while removing the negative sign in the exponent.

Quotient of Powers Rule

The Quotient of Powers Rule helps simplify expressions where the same base is involved in a division. This rule is handy in algebra and essential when working with exponents. According to the Quotient of Powers Rule, you subtract the exponent of the denominator from the exponent of the numerator when the bases are the same.
Given the general rule:

• \( \frac{b^m}{b^n} = b^{m-n} \)

Think of it as simplifying how the bases are distributed across a fraction. For example, in the expression \( \frac{a^{-2}}{a^2} \), the base \( a \) remains the same, and subtracting the exponents \( -2 \) and \( 2 \) gives you \( -4 \). This results in \( a^{-4} \). This step simplifies everything into one expression with a single exponent.

Reciprocal

Understanding reciprocals is a fundamental aspect of working with negative exponents. Reciprocals involve flipping a number, so the numerator becomes the denominator and vice versa. This concept is incredibly useful when handling problems where negative exponents are present.
Suppose you have \( a^{-n} \). The task is to convert it into its reciprocal form which looks like \( \frac{1}{a^n} \).

• Turning \( a^{-n} \) into \( \frac{1}{a^n} \) is the reciprocal transformation.
• It's essential because it changes negative exponents into positive ones while keeping the value of the expression consistent.

By converting \( a^{-4} \) into \( \frac{1}{a^4} \), we ensure the final expression contains no negative exponents, aligning it with mathematical conventions for expressing final results.