Problem 68 - Elementary and Intermediate Algebra

Question

Simplify. Do not use negative exponents in the answer. \(\left(n^{3}\right)^{-5}\)

Step-by-Step Solution

Answer

The simplified form is \(\frac{1}{n^{15}}\).

Step 1: Understand Negative Exponents

A negative exponent, such as \(a^{-n}\), means that you take the reciprocal of the base \(a\) raised to the positive exponent. So, \(a^{-n} = rac{1}{a^n}\).

Step 2: Apply Negative Exponent Rule

In \((n^3)^{-5}\), the base is \(n^3\) and the exponent is \(-5\). According to the rule for negative exponents, this expression can be rewritten as \(\frac{1}{(n^3)^5}\).

Step 3: Apply Power of a Power Rule

The power of a power rule states that \((a^m)^n = a^{m \cdot n}\). Therefore, apply this rule to take the expression \(\frac{1}{(n^3)^5}\) to \(\frac{1}{n^{3 \times 5}} = \frac{1}{n^{15}}\).

Step 4: Final Simplified Form

The expression is now fully simplified and written without negative exponents as \(\frac{1}{n^{15}}\).

Key Concepts

Negative Exponents

Negative exponents can seem tricky at first, but they're quite straightforward once you know the rule. If you see an expression with a negative exponent, it's necessary to change it to its reciprocal with a positive exponent. Consider the expression \(a^{-n}\). This simply means "one over \(a\) raised to the \(n\)th power." To put it another way: \(a^{-n} = \frac{1}{a^n}\). This transformation is crucial because it simplifies expressions and avoids negative exponents, which are often difficult to interpret directly. For example, in our original expression \((n^3)^{-5}\), we convert it to \(\frac{1}{(n^3)^5}\) using this rule. This initial step lays the foundation for further simplifications.

Power of a Power Rule

The power of a power rule is an essential tool in dealing with expressions raised to another power. When you have an exponent raised to another exponent, like \((a^m)^n\), you can simplify it by multiplying the exponents together. This results in \(a^{m \cdot n}\). In our problem, we have \((n^3)^5\). Applying the power of a power rule here, we multiply the exponents: \(3 \times 5 = 15\). Therefore, \((n^3)^5 = n^{15}\). This step takes our expression to \(\frac{1}{n^{15}}\). The power of a power rule turns nested expressions into a more direct form, simplifying calculations and improving readability.

Simplifying Expressions

Simplifying expressions is the process of making them as concise and clear as possible. In mathematics, simplifying often involves reducing the number of operations and making the expression easier to work with. After using the negative exponent rule and the power of a power rule, our expression \((n^3)^{-5}\) transforms into \(\frac{1}{n^{15}}\). This is a simpler and more straightforward form. Rundown of the steps includes:

• Using the negative exponent rule to change \((n^3)^{-5}\) to \(\frac{1}{(n^3)^5}\).
• Applying the power of a power rule to simplify \((n^3)^5\) to \(n^{15}\).
• Converting the expression into \(\frac{1}{n^{15}}\), the final simplified form with only positive exponents.

By completing these steps, you arrive at a much simpler expression, making calculations and understanding a lot easier.