Problem 62 - Algebra 1: Concepts and Skills

Question

Simplify the expression. The simplified expression should have no negative exponents. (Lesson 8.4). $$ \frac{\left(a^{3}\right)^{4}}{\left(a^{3}\right)^{8}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{1}{a^{12}} \).

Step 1: Apply power-of-a-power rule

To simplify the given expression, we first apply the power-of-a-power rule for both the numerator and the denominator. This means we multiply the exponents in both the numerator and the denominator. The expression \(\left(a^{3}\right)^{4}\) in the numerator simplifies to \(a^{3*4} = a^{12}\). Similarly, in the denominator, the expression \(\left(a^{3}\right)^{8}\) simplifies to \(a^{3*8} = a^{24}\). So, now we have \(\frac{a^{12}}{a^{24}}\).

Step 2: Apply division rule

Now, apply the division rule. This rule states that when we divide with the same base, we subtract the exponents. Hence, we subtract the exponent in the denominator from the one in the numerator. That gives us \(a^{12-24} = a^{-12}\).

Step 3: Handle negative exponent

We are asked to present the answer without negative exponents. Remember that a negative exponent simply means take the reciprocal. So, \(a^{-m} = \frac{1}{a^m}\). Hence, \(a^{-12} = \frac{1}{a^{12}}\). This is the final answer.

Key Concepts

Negative Exponents

When working with exponents, encountering negative exponents might initially seem confusing, but they're just a logical extension of rules you already know. A negative exponent indicates that you should take the reciprocal of the base raised to the corresponding positive exponent.
For example, let's consider the expression \(a^{-m}\). This can be rewritten as:

• \(a^{-m} = \frac{1}{a^m}\)

The rule makes it clear that instead of multiplying the base by itself a negative number of times, which doesn't make sense, you take the reciprocal instead.
So, if you have \(a^{-12}\), you should express it as \(\frac{1}{a^{12}}\). This makes the expression positive, simply by flipping the base to the denominator.

Power-of-a-Power Rule

The power-of-a-power rule is a handy shortcut that saves you time and effort when simplifying expressions with multiple exponents. This rule states that if you raise a power to another power, simply multiply the exponents together.
Here's how it works:

• If you have \((a^m)^n\), it simplifies to \(a^{m \times n}\).

In our original expression, both the numerator and denominator had powers raised to another power. For example, the numerator \((a^3)^4\) became \(a^{3 \times 4} = a^{12}\). Similarly, in the denominator, \((a^3)^8\) became \(a^{3 \times 8} = a^{24}\).
Always remember this rule when faced with nested exponents. It can significantly simplify expressions, making the problem-solving process much smoother.

Division Rule

Simplifying expressions often involves division, and applying the division rule with exponents is straightforward once you grasp the concept. When dividing expressions with the same base, the division rule tells us to subtract the exponent in the denominator from the exponent in the numerator. This helpful trick will simplify fractions involving similar bases raised to different powers.
The basic form of this rule is:

• \(\frac{a^m}{a^n} = a^{m-n}\).

In the initial exercise, after applying the power-of-a-power rule, we had \(\frac{a^{12}}{a^{24}}\). By applying the division rule, we subtract 24 from 12, resulting in \(a^{12-24} = a^{-12}\).
This step helps reduce complexity and is crucial for moving expressions closer towards their simplest form. Understandably, using these rules in tandem gets you the answer efficiently, without much computational effort.