Problem 71 - Algebra 1

Question

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

Step-by-Step Solution

Answer

The expression \(\frac{1}{2 x^{-5}}\) rewritten with positive exponents is \(0.5x^5.\)

Step 1: Identify the base and the exponent

In the given expression, the base is 'x' and its exponent is '-5'. Also, there's a fraction where \(2x^{-5}\) is in the denominator.

Step 2: Apply the negative exponent rule

According to the rule of negative exponents, \(x^{-5} = 1/x^5\). Apply this rule to transform the expression with negative exponents to positive exponents.

Step 3: Rewrite the expression

Rewrite the fraction with the positive exponent now. Thus, the expression \(\frac{1}{2x^{-5}}\) becomes \(\frac{1}{2} * x^5\) which is equivalent to \(0.5x^5.\).

Key Concepts

Negative Exponent Rule

The negative exponent rule is a fundamental concept in algebra. It helps us to understand how to handle bases with negative exponents. The rule states that any base raised to a negative exponent is equivalent to its reciprocal raised to the positive of that exponent. For example, if you have an expression like \(x^{-n}\), it can be rewritten as \(\frac{1}{x^n}\). This means you "flip" the base to the denominator and change the exponent's sign.
When working with negative exponents:

• Identify the base with the negative exponent.
• Rewrite it as a fraction by placing the base in the denominator.
• Convert the exponent to a positive number.

This rule is an effective tool in algebra, simplifying expressions and making them easier to work with.

Expression Simplification

Expression simplification involves rewriting complex mathematical expressions in a simpler form. This process usually involves reducing fractions, combining like terms, or applying various algebraic rules like the negative exponent rule.
Simplifying expressions:

• Makes the expression easier to understand and use.
• Is crucial in solving algebraic equations efficiently.
• Helps avoid arithmetic mishaps during calculations.

In the example given, simplifying the expression involved applying the negative exponent rule to remove a negative exponent, and then rewriting the entire expression without fractions, culminating in a simpler solution: \(0.5x^5\). By simplifying the expression, we not only transform it to a neater form but also retain its original value in a more usable format.

Algebraic Fractions

Algebraic fractions are essentially fractions where the numerator, the denominator, or both contain algebraic expressions. Just like normal fractions, algebraic fractions can be simplified by factoring and reducing wherever possible.
Working with algebraic fractions includes:

• Identifying common factors in the numerator and the denominator.
• Cancelling out these common factors to simplify the fraction.

Additionally, managing negative exponents in algebraic fractions can seem tricky, but using rules like the negative exponent rule allows us to rewrite fractions in a more straightforward form. In our problem, the denominator contained a negative exponent, \(x^{-5}\), which we rewrote using the rule to simplify the fraction. Instead of remaining as a complex denominator, it becomes part of the simplified expression, turning \(\frac{1}{2x^{-5}}\) into \(0.5x^5\). This transformation is key in handling algebraic expressions cleanly and efficiently.