Problem 18 - Elementary Algebra

Question

When simplifying the terms for the following problems, write each so that only positive exponents appear. $$ \left(a^{4}\right)^{-3} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\left(a^{4}\right)^{-3}\) and ensure that only positive exponents appear in the simplified expression. Answer: \(\frac{1}{a^{12}}\)

Step 1: Apply the rule of exponents (Power of a Power)

According to the rule of exponents, when we have a power raised to another power, we have to multiply the exponents. In this case, we have \(\left(a^{4}\right)^{-3}\), so we need to multiply the exponents 4 and -3 to simplify the expression: $$ \left(a^{4}\right)^{-3} = a^{4 \times (-3)} $$

Step 2: Multiply the exponents and rewrite the expression with a positive exponent

Now, we need to multiply 4 by -3 to get the actual exponent: $$ a^{4 \times (-3)} = a^{-12} $$ However, the expression still has a negative exponent, and we must change it to a positive exponent. Using the rule of exponents for negative exponents, we know that \(a^{-n} = \frac{1}{a^n}\). So, we can rewrite the expression as: $$ a^{-12} = \frac{1}{a^{12}} $$

Step 3: Write the final simplified expression

Now that we have the expression with only positive exponents, we can write the final simplified expression as: $$ \left(a^{4}\right)^{-3} = \frac{1}{a^{12}} $$

Key Concepts

Power of a Power Rule

Understanding the power of a power rule is fundamental for simplifying expressions with exponents. It's quite straightforward: when you have an exponent raised to another exponent, you multiply the two exponents. For instance, consider the expression \( (a^{m})^{n} \). Here, you simply multiply the exponents \(m\) and \(n\) to simplify the expression, resulting in \(a^{mn}\).

Let's take our textbook exercise \( (a^{4})^{-3} \) as an example. We apply the rule by multiplying 4 and -3, giving us \( a^{4 \times (-3)} \), which simplifies further to \( a^{-12} \). While this step is crucial, we also need to ensure the result has positive exponents, which leads us to the next concept: dealing with negative exponents.

Negative Exponents

Negative exponents often confuse students, but they follow a simple principle that turns them into positive ones. The rule states that any term with a negative exponent can be rewritten as its reciprocal with a positive exponent. In mathematical terms, \( a^{-n} \) becomes \( \frac{1}{a^{n}} \).

Looking back at our problem where we ended up with \( a^{-12} \), applying this rule we convert it to \( \frac{1}{a^{12}} \). Always remember, the negative exponent indicates that the base, in this case \(a\), is on the opposite side of the fraction than it originally appeared. By understanding this concept, you ensure that your final simplified expression will only feature positive exponents.

Exponent Multiplication

When simplifying expressions that involve multiplying exponents with the same base, we add the exponents according to the exponent multiplication rule. For example, when you have \( a^{m} \times a^{n} \), the simplified form is \( a^{m+n} \). This rule is essential when combining multiple exponential terms.

However, it's important to note that this rule is only applicable when the bases are the same and when we multiply, not when we raise an exponent to another exponent—that's when we use the power of a power rule. In our previous examples, we focused on the latter, but this multiplication rule is equally vital in understanding the broader scope of simplifying exponential expressions.