Problem 45 - Algebra 1

Question

Rewrite the expression with positive exponents $$\left(\frac{-4 x^{2}}{2 x^{-1}}\right)^{-1}$$

Step-by-Step Solution

Answer

The expression with positive exponents is \(-(1/(2x^{3}))\).

Step 1: Apply the power rule

Here we have an expression raised to the power -1. By the power rule of exponents, \((-4x^2/(2x^{-1}))^{-1}\) turns into \(1/(-4x^2/(2x^{-1}))\).

Step 2: Simplify the fraction inside the brackets

To simplify \(-4x^2/2x^{-1}\), we convert the exponent of \(x\) to positive, which gives us \(-4x^2/(2/x)\). By multiplying the numerator and denominator with the reciprocal of the fraction in the denominator we get, this simplifies to \(-4x^{3}/2 = -2x^{3}\). So the expression becomes \(1/(-2x^{3})\).

Step 3: Simplify the expression

Since the fraction \(1/(-2x^{3})\) does not need any more simplification, this is our final result. Hence, the expression with positive exponents is \(-(1/(2x^{3}))\).

Key Concepts

Power Rule of Exponents

The power rule of exponents is a fundamental concept in algebra that simplifies the process of raising powers to another power. It states that when you have a power raised to another power, you multiply the exponents. For instance, \( (x^m)^n = x^{mn} \). This rule is extremely handy when simplifying exponential expressions.

When looking at the problem \( (-4x^2 / 2x^{-1})^{-1} \), it can be daunting at first. However, by applying the power rule, you reverse the scenario. Now the entire fraction is in the denominator, flipped as \( 1/(-4x^2 / 2x^{-1}) \). This inversion due to the exponent of -1 demonstrates the elegance of the power rule in transforming expressions, which is your first step in simplification.

Negative Exponents

Dealing with negative exponents can seem tricky, but they follow a simple rule: having a negative exponent means you take the reciprocal of the base and then apply the positive exponent. In essence, \( x^{-n} = 1/x^n \) (for \( x \eq 0 \) )

Let's apply this when simplifying the fraction inside the expression. The term \( x^{-1} \) can be rewritten as \( 1/x \) which turns our expression \( -4x^2/2x^{-1} \) into \( -4x^2/(2/x) = -4x^2 \cdot \frac{x}{2} = -4x^3/2 \). Understanding and applying the concept of negative exponents is crucial, as it allows you to simplify expressions by converting all exponents to positive, making it easier to work with them.

Fraction Simplification

Fraction simplification is about making expressions more manageable and easier to understand by breaking down complex fractions into simpler components. This often includes reducing the numerator and the denominator to their smallest possible values. In algebra, simplifying expressions might also involve getting rid of negative exponents and simplifying any algebraic terms.

In the given exercise, after applying previous rules, we have the term \( -4x^3/2 \). Simplifying the fraction further, we notice that both numerator and denominator are divisible by 2, allowing us to simplify the expression to \( -2x^3 \), effectively tidying up the overall expression. With fraction simplification, clarity becomes the focus, turning potentially complex fractions into ones that are far more straightforward and easier to work with.

This straightforwardness is exemplified in our final result \( -(1/(2x^{3})) \), a much cleaner version of our original complex problem.