Problem 26 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. $$ \frac{1}{10^{-1}} $$

Step-by-Step Solution

Answer

The expression \(\frac{1}{10^{-1}}\) equals 10

Step 1: Understanding Negative Exponents

A number raised to the power of -1 (or any negative power), like \(10^{-1}\), is just the reciprocal of that number raised to the positive power, or in this case \(1\div10^1\).

Step 2: Evaluate \(10^{-1}\)

Applying the rule described above, \(10^{-1}\) equals \(1\div10^1\), or \(1\div10\), which is 0.1.

Step 3: Evaluate \(1\div 10^{-1}\)

Substitution leads to \(1\div0.1\). Dividing 1 by 0.1 equals 10.

Key Concepts

Understanding Negative Exponents

Negative exponents often cause confusion, but they can be understood as instructions to take the reciprocal of a number. If you encounter an expression like \( a^{-n} \), what this really means is that you should take the reciprocal of \( a \) to the \( n \)th power. This results in \( \frac{1}{a^n} \).

For example, to evaluate an expression with a negative exponent such as \( 10^{-1} \), you would consider the reciprocal of 10 to the positive power of 1. In this case, you're looking at \( \frac{1}{10^1} \), which simplifies to \( \frac{1}{10} \). As a decimal, this represents 0.1. This simple switch—converting a negative exponent to a reciprocal—is the key to clarifying what may initially seem like a complex concept.

Reciprocal of Exponents

The reciprocal of an exponent refers to the inverse operation concerning exponents. If a number is raised to a power, the reciprocal of this is the number raised to the negative of that power. It's important to note that the reciprocal of an exponent flips the base number; if the base is greater than one, the reciprocal will be less than one, and vice versa.

For instance, when dealing with the expression \( 10^{-1} \), the reciprocal of \( 10^1 \) is sought. In simpler terms, determine what number multiplied by \( 10^1 \) equals 1. The answer, consistent with the definition of \( b^{-n} = \frac{1}{b^n} \), is 0.1, which is precisely \( \frac{1}{10} \). Visualizing this relationship between positive and negative exponents helps make handling them in more complicated algebraic expressions much more manageable.

Simplifying Algebraic Expressions

Simplifying algebraic expressions with negative exponents involves recognizing patterns and applying exponent rules. The overarching goal is to rewrite the expression in its simplest form, which usually means with positive exponents and without fractions, if possible.

To simplify an expression like \( \frac{1}{10^{-1}} \), you must recognize that the presence of a negative exponent calls for using the reciprocal. This turns the expression into \( 1 \times 10 \), since the reciprocal of \( 10^{-1} \) is \( 10 \). Now the expression is devoid of negative exponents and easy to compute. The final simplified result of this particular expression is 10, much clearer than the initial fraction form.

Overall, by understanding how to manipulate exponents, specifically switching between negative exponents and their reciprocal equivalents, simplifying even the most daunting algebraic expressions can be achieved with ease.