Problem 115 - Elementary Algebra

Question

For the following problems, evaluate each numerical expression. $$ (-10)^{-1} $$

Step-by-Step Solution

Answer

Answer: \(\frac{-1}{10}\)

Step 1: Understanding Negative Exponents

A negative exponent, such as -1 in this case, indicates that we take the reciprocal of the base. In other words, the expression \((-10)^{-1}\) is equivalent to \(\frac{1}{(-10)^1}\).

Step 2: Evaluate the Expression

Since we have now converted the expression to a fraction with a positive exponent, we can evaluate the expression. We know that \((-10)^1 = -10\), so our expression becomes: $$ \frac{1}{(-10)^1} = \frac{1}{-10} $$ Since \(1\) divided by \(-10\) is simply \(\frac{-1}{10}\), the expression \((-10)^{-1}\) evaluates to \(\frac{-1}{10}\).

Key Concepts

Negative Exponent Rules

When you come across a negative exponent, it might seem a bit confusing at first. But don't worry, the rules are simple and consistent. A negative exponent tells us to do the opposite of multiplying—it advises us to divide. More specifically, a number raised to a negative exponent is equal to the reciprocal of that number raised to the opposite positive exponent.

Take the expression \( (-10)^{-1} \). According to the negative exponent rule, we flip the base and make the exponent positive. This flips our expression to \( \frac{1}{(-10)^1} \), where \( (-10)^1 \) is just -10. So, why does this rule exist? It's because exponents are fundamentally about multiplication, and a negative exponent symbolizes division, or multiplying by the reciprocal.

Reciprocal of a Number

The reciprocal of a number is what you get when you divide 1 by that number. It’s like flipping a fraction upside down—if you have a fraction \( \frac{a}{b} \) its reciprocal would be \( \frac{b}{a} \). For whole numbers or integers, we usually express the reciprocal as a fraction with the number in the denominator.

Let's say we want to find the reciprocal of -10. We write 1 divided by -10, which is \( \frac{1}{-10} \) or -0.1. Reciprocals are essential when working with negative exponents because they convert the problem into a form that is often easier to understand and solve, as seen in the original exercise solution.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a fundamental skill in mathematics. It involves combining like terms, reducing fractions, and applying exponent rules to make expressions more manageable. The goal is to make the expression as simple as possible without changing its value.

Take the example of our negative exponent expression \( (-10)^{-1} \). It initially looks complicated due to the negative exponent, but by applying the rule for negative exponents, we simplify it to \( \frac{1}{-10} \). Simplification is about making sure that the expressions are in their most reduced form, which makes them easier to read and work with. In all cases, you might need to apply several rules, such as distributive properties, combining like terms, or canceling out common factors in numerators and denominators.