Problem 51 - Algebra 1: Concepts and Skills

Question

Rewrite the expression with positive exponents. $$ x^{-5} $$

Step-by-Step Solution

Answer

The expression \(x^{-5}\) with positive exponents is \(\frac{1}{x^5}\)

Step 1: Apply the negative exponent rule

By definition, \(a^{-n} = \frac{1}{a^n}\) for \(a \neq 0\).

Step 2: Rewrite

\(x^{-5} = \frac{1}{x^5}\)

Key Concepts

Algebraic Expressions and Negative Exponents

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators. They are fundamental components of algebra that students encounter in various forms. A particularly intriguing yet sometimes perplexing concept within algebraic expressions is the use of negative exponents. A negative exponent indicates that the base (the number being raised to a power) is on the wrong side of a fraction and needs to be inverted.

For example, in the given expression \(x^{-5}\), \(x\) is the base and \(5\) is the negative exponent. To convert this to an expression with a positive exponent, we can use the key principle that \(x^{-n} = \frac{1}{x^n}\). So, by applying this rule, \(x^{-5}\) becomes \(\frac{1}{x^5}\), which is much easier to work with in algebraic operations. Such transformations are crucial for simplifying expressions and solving equations.

Understanding Exponent Rules

Exponent rules, also known as laws of exponents, are essential guidelines that help simplify expressions involving powers. One of these fundamental rules is the handling of negative exponents, which is vital to correctly simplifying algebraic expressions. The negative exponent rule directly converts expressions with negative exponents to their positive counterparts leading to a cleaner and more consistent expression.

This rule is pretty straightforward: for any nonzero base \(x\) and positive integer \(n\), \(x^{-n} = \frac{1}{x^n}\). The negative sign in the exponent indicates that you should take the reciprocal of the base raised to the positive exponent. Keeping this rule in mind makes it much easier to deal with complex algebraic terms that might otherwise be intimidating.

Simplifying Expressions with Negative Exponents

Simplifying expressions is a key skill in algebra that involves rewriting expressions in an equivalent, but typically more convenient, form. When dealing with negative exponents, simplification means transforming the expression into one where all exponents are positive. The crux of this process lies in understanding and applying the rules of exponents in a methodical way.

By acknowledging that a negative exponent represents the reciprocal of the base raised to the opposing positive exponent, students can systematically approach the simplification of algebraic expressions. In the exercise \(x^{-5}\), simplifying the expression is as simple as applying the rule and rewriting it as \(\frac{1}{x^5}\). Remember, the goal of simplification is to ensure that the expressions are easy to read and ready for further manipulation, which in turn helps with understanding the overall problem and finding a solution.