Problem 6 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. $$ \frac{1}{(-2)^{-1}} $$

Step-by-Step Solution

Answer

-2

Step 1: Understand Negative Exponents

A negative exponent means the reciprocal of the base. -2 raised to the power of -1 is equivalent to \(\frac{1}{-2}\).

Step 2: Solve the Exponent Part

Calculate \((-2)^{-1}\), which equals \(-0.5\).

Step 3: Evaluate the Final Expression

The initial expression now becomes \(\frac{1}{-0.5}\). This division calculates to \(-2\).

Key Concepts

Negative Exponents

Grasping the concept of negative exponents plays a crucial role in understanding more advanced mathematical operations. When we talk about negative exponents, we're essentially considering the inverse procedure to usual exponentiation. For instance, when you encounter an expression like \( a^{-n} \), it's not meant to perplex you with negativity, but rather to cue you in on a simple transformation - switching to the reciprocal form. In mathematical terms, this means \( a^{-n} = \frac{1}{a^{n}} \).

This reciprocal change flips the base and elevates it to the positive exponent. If we apply this to our example, \( (-2)^{-1} \) would turn into \( \frac{1}{(-2)^1} \), which simplifies to \( \frac{1}{-2} \). When digesting negative exponents, remember it's just a way to express division by a number's positive exponentiated form.

Reciprocal of the Base

The reciprocal of a number is a flipped version of it, mathematically speaking. To find the reciprocal, you simply divide 1 by that number. It's like looking at the world through the other side of the mirror. In arithmetic, every number's reciprocal is unique and serves as a sort of mathematical twin. Notably, the reciprocal of a base with a negative exponent is related to this concept. If the base is \( a \), the reciprocal is \( \frac{1}{a} \), and if we apply a negative exponent, the relationship is just as significant: \( a^{-1} = \frac{1}{a} \).

With the example provided, \( (-2)^{-1} \) gives us \( \frac{1}{-2} \) indeed. This is because taking the reciprocal here is akin to attributing a negative one as the exponent, which is just another way to express that the base is being inverted.

Simplifying Expressions

Simplifying expressions can sometimes feel like tidying up a messy room, making everything neat and as straightforward as possible. With mathematical expressions, we aim to rewrite them in their simplest form, removing any unnecessary complexity. This often involves following the rules of exponents, like those for negative exponents, which clear the way for easier calculation.

When simplifying an expressions like \( \frac{1}{(-2)^{-1}} \), actions speak louder than mere observation. You first address the exponent, transforming \( (-2)^{-1} \) into \( -0.5 \), then revisit the whole expression. Now your task turns to dividing 1 by -0.5, which perfectly streamlines to \( -2 \). Simplifying is all about making life easier, breaking down seemingly complex tasks into manageable, bite-sized pieces.