Problem 30 - Algebra 1

Question

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

Step-by-Step Solution

Answer

The expression \(x^{-5}\) rewritten with a positive exponent is \(1/x^5\).

Step 1: Express the Exponential Term as a Fraction

Given the expression \(x^{-5}\), the first step is to rewrite it in fraction form according to the law of exponents. The expression can be rewritten as \(1/x^5\). This is because \(x^{-5} = 1/x^5\).

Step 2: Simplify

In this step, you will notice that there are no further operations that can be performed on the expression, so \(1/x^5\) is the simplified version of \(x^{-5}\) with a positive exponent.

Key Concepts

Understanding Negative Exponents

Negative exponents can seem tricky at first, but they're quite simple after a brief explanation. A negative exponent indicates that the base should be taken as a reciprocal. For instance, in the expression \(x^{-5}\), the negative exponent \(-5\) tells us to "flip" the base. Instead of making calculations more complex, negative exponents streamline mathematical expressions.

• Negative exponents are essentially a way to represent division in powers.
• For any non-zero number \(a\), \(a^{-n} = \frac{1}{a^n}\).
• This means \(x^{-5}\) becomes \(\frac{1}{x^5}\).

Understanding how to handle them will help simplify and solve exponent-based problems more easily.

Law of Exponents

The laws of exponents give us the rules to manipulate expressions involving powers. This comes in handy, especially when dealing with expressions that have negative exponents.
Here's a quick overview of some key laws:

• Product of Powers: \(a^m \times a^n = a^{m+n}\)
• Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power of a Power: \((a^m)^n = a^{m \cdot n}\)
• Negative Exponent: \(a^{-n} = \frac{1}{a^n}\)

These laws are like the grammar rules of exponents. By applying the rule for negative exponents, like in our example \(x^{-5} = \frac{1}{x^5}\), we rearrange the expression so the exponent is positive, simplifying further calculations.

Simplification of Expressions

Simplifying expressions is the process of making them easier to work with and understand. With exponents, this means using the laws of exponents to rewrite them in a simpler form.

• Start by applying the law of exponents that fits the situation. For \(x^{-5}\), use the negative exponent rule.
• Turn negative exponents into positive by taking the reciprocal of the base and converting the exponent.
• Simplified expressions are usually easier to plug into further calculations or equations.

In our example, \(x^{-5}\) simplifies to \(\frac{1}{x^5}\). The steps are straightforward, leaving us with the simplest form of the expression. This enables more efficient problem-solving in algebra and beyond.