Question

Rewrite the expression with positive exponents. $$\frac{1}{2 x^{-5}}$$

Short answer

The simplified expression with positive exponents is \(\frac{2}{x^5}\).

Step-by-step solution

Rewrite the Reciprocal

Rewriting the reciprocal, the given expression can be written as \(2x^{-5}\).

Handle the Negative Exponent

By definition, we know that \(x^{-n}=\frac{1}{x^n}\). According to this definition, \(x^{-5}=\frac{1}{x^5}\) . So the expression can be rewritten as \(2 \times \frac{1}{x^5}\).

Simplify the Expression

The expression now simplifies to \(\frac{2}{x^5}\)

Key concepts

Simplifying Expressions

In algebra, simplifying expressions is the process of making them easier to understand or work with. This can include condensing multiple steps into one, reducing fractions, or eliminating negative exponents as we see in the provided exercise. Simplification often involves recognizing and applying basic mathematical principles, like the properties of exponents.

For example, understanding how to work with negative exponents is crucial for simplification. It’s a common misconception that a negative exponent means a negative number, but it actually means that the base (in this case, our variable 'x') needs to be on the bottom of a fraction, as the opposite of a positive exponent, which places the base on the top.

To simplify the expression \(\frac{1}{2x^{-5}}\), we need to apply the negative exponent rule, which states that \(a^{-n} = \frac{1}{a^n}\). When we convert \(x^{-5}\) into \(\frac{1}{x^5}\), we're simplifying the expression to make it clearer and easier to work with, resulting in the final answer \(\frac{2}{x^5}\).

Reciprocal of a Number

The reciprocal of a number is simply a flipped version of the original number—it's what you get when you divide 1 by that number. For a non-zero number 'a', the reciprocal is \(\frac{1}{a}\) or \(a^{-1}\). The concept of finding a reciprocal becomes particularly handy when dealing with negative exponents.

When you have a term like \(x^{-5}\), you need to remember that the negative sign in the exponent indicates the reciprocal. So, the reciprocal of \(x^5\) is \(x^{-5}\), and accordingly, \(x^{-5} = \frac{1}{x^5}\). This simple reciprocal flip helps transform our original exercise from a negative exponent to a positive one, allowing us to simplify it further.

Algebraic Manipulation

Algebraic manipulation is the process of changing the form of an algebraic expression without changing its value. This skill involves a combination of techniques, including factoring, expanding, and using properties of operations such as the distributive property.

In our original exercise, we saw algebraic manipulation at play when the negative exponent was handled. Recognizing that \(x^{-5}\) is equivalent to \(\frac{1}{x^5}\), we performed an operation to rewrite the expression. This is a clear example of manipulating the form of the expression (from a negative to a positive exponent) while maintaining its mathematical integrity.

Through these manipulations, we can often make calculations easier, prepare expressions for solving equations or inequalities, or simply make the expressions clearer for further mathematical operations.