Problem 38 - Intermediate Algebra

Question

Write with positive exponents. Simplify if possible. $$ \frac{1}{n^{-k 9}} $$

Step-by-Step Solution

Answer

\(n^{k9}\)

Step 1: Understand Negative Exponents

The negative exponent indicates that the base, here represented by \(n^{-k9}\), is in the denominator and can be rewritten as a reciprocal. The rule for negative exponents states \(a^{-m} = \frac{1}{a^m}\). Conversely, \(\frac{1}{a^{-m}} = a^m\).

Step 2: Apply the Rule for Negative Exponents

By applying the rule from step 1, the expression \(\frac{1}{n^{-k9}}\) can be rewritten as \(n^{k9}\) because the negative exponent turns into a positive when moved from the denominator to the numerator.

Key Concepts

Negative Exponents

Negative exponents are quite an interesting topic in mathematics. Simply put, a negative exponent tells us to take the reciprocal of the base and then raise it to the absolute value of the exponent. So, when you see something like \(n^{-k9}\), it means you have to "flip" the base \(n\). Hence, \(n^{-k9}\) becomes \(\frac{1}{n^{k9}}\).

In a way, negative exponents can be thought of as a shortcut for creating reciprocals.

• Negative exponents indicate the inverse of a power.
• They simplify expressions and keep them neatly organized.

This makes them a helpful tool when working with fractions and division in algebra.

Reciprocal Rule

The reciprocal rule is key to understanding negative exponents. Whenever you encounter a fraction with a negative exponent in the denominator, such as \(\frac{1}{n^{-k9}}\), you can use the reciprocal rule.

• The rule states that any number to a negative exponent equals the reciprocal of that number to the opposite positive exponent.
• Apply this by flipping the fraction and changing the sign of the exponent to positive.

As a result, using the reciprocal rule simplifies \(\frac{1}{n^{-k9}}\) to \(n^{k9}\).
Remember, this is an incredibly handy rule for simplifying algebraic expressions, especially when you wish to express them without fractions.

Algebraic Expressions

Algebraic expressions can become complex, but understanding concepts like negative exponents and the reciprocal rule can greatly simplify them. These expressions involve numbers, variables, and operations. In the expression \(n^{-k9}\), \(n\) is a variable and \(-k9\) is the exponent attached to it.

• Simplifying algebraic expressions often requires using exponent rules.
• Expressing exponents positively makes the expression easier to manage.

By turning \(\frac{1}{n^{-k9}}\) into \(n^{k9}\), the expression is simplified, allowing for more straightforward calculations and interpretation.

Understanding how to manipulate these pieces enhances problem-solving skills and prepares you for more complex mathematics.