Problem 36 - Algebra 1

Question

Simplify the expression. The simplified expression should have no negative exponents. $$ \left(\frac{a^{9}}{a^{5}}\right)^{-1} $$

Step-by-Step Solution

Answer

The simplified expression of \(\left(\frac{a^{9}}{a^{5}}\right)^{-1}\) is \( \frac{1}{a^4}\) .

Step 1: Understand the expression

The provided expression is \(\left(\frac{a^{9}}{a^{5}}\right)^{-1}\). Applying the rule -- when dividing numbers with the same base, subtract their exponents-- helps to simplify the expression within the parentheses.

Step 2: Simplify the base

Apply the law of exponents and subtract 5 from 9 in the exponent, which gives \(a^{9-5} = a^4\). So, the expression becomes \( (a^{4})^{-1} \) .

Step 3: Address the negative exponent

A negative exponent simply means we find the reciprocal of the base. Hence, \( (a^{4})^{-1} = \frac{1}{a^4} \).

Key Concepts

Negative Exponents

Negative exponents can sometimes be confusing, but they carry an important and simple meaning. When you see a negative exponent, it tells you to flip the number, or variable, to its reciprocal form. More formally, if you have an expression like \(a^{-n}\), it can be rewritten as \(\frac{1}{a^n}\). This means the negative sign in the exponent directs you to "invert" the base to its reciprocal before considering the power.

This concept is crucial when simplifying expressions that feature negative exponents. You essentially remove the negative sign by flipping the base and then apply any other operations that are necessary, such as raising it to the power that's specified.

Exponent Rules

When working with exponents, there are several rules that can help make your calculations straightforward. These rules are born from mathematical logic, and they apply to problems involving powers and bases. Here are a few key ones to keep in mind:

• Product of Powers Rule: \(a^m \cdot a^n = a^{m+n}\) - When multiplying like bases, add their exponents.
• Quotient of Powers Rule: \(\frac{a^m}{a^n} = a^{m-n}\) - When dividing like bases, subtract the exponents.
• Power of a Power Rule: \((a^m)^n = a^{m\cdot n}\) - When raising a power to another power, multiply the exponents.

In the expression \(\left(\frac{a^9}{a^5}\right)^{-1}\), these rules apply. The Quotient Rule helps simplify \(a^{9-5} = a^4\). Maintaining a clear understanding of these rules assists in tackling many exponent-based problems smoothly.

Reciprocal

The term "reciprocal" refers to the inverse of a number or expression. To find the reciprocal of a number, you simply flip it around. For example, the reciprocal of \(a\) is \(\frac{1}{a}\) and vice versa. When it comes to expressions with variables raised to exponents, like \(a^n\), finding the reciprocal involves the entire expression, turning it into \(\frac{1}{a^n}\).

Reciprocals are especially handy when dealing with negative exponents. As we saw in the exercise \((a^{4})^{-1}\), instead of \(a^4\) being in the numerator, its reciprocal \(\frac{1}{a^4}\) places it in the denominator.
Understanding how to find reciprocals is a key skill, particularly useful in algebra and complex mathematical manipulations where inverse relationships and simplifications are required.