Problem 37 - Intermediate Algebra

Question

Write with positive exponents. Simplify if possible. $$ \frac{1}{a^{-2 / 3}} $$

Step-by-Step Solution

Answer

The expression is rewritten as \( a^{2/3} \).

Step 1: Identify the Expression

The exercise requires us to rewrite \( \frac{1}{a^{-2/3}} \) with positive exponents. Currently, we have a negative exponent in the denominator.

Step 2: Use the Negative Exponent Rule

The negative exponent rule states that \( a^{-n} = \frac{1}{a^{n}} \). Applying this rule to the denominator, \( a^{-2/3} \), we can rewrite the fraction as \( 1 \times a^{2/3} \).

Step 3: Simplify the Expression

Simplifying, we get \( a^{2/3} \), which is the expression with a positive exponent and is already simplified.

Key Concepts

Negative Exponent Rule

When you see an expression with a negative exponent, it might seem a bit daunting at first. But fear not! There is a straightforward rule that helps us deal with such cases. It's called the **Negative Exponent Rule**. This rule tells us that a negative exponent indicates that the base should be taken as a reciprocal and rewritten with a positive exponent.
To break it down, here's how it works:

• Given a term \( a^{-n} \), the negative exponent rule states \( a^{-n} = \frac{1}{a^{n}} \).
• This means that the term is equivalent to the reciprocal of the base raised to a positive exponent.

This is how we transformed \( \frac{1}{a^{-2/3}} \) to \( a^{2/3} \) in the original exercise. By switching the position of the base from the denominator layer to the numerator, we change the sign of the exponent from negative to positive. This makes it easier to read and simplify.

Simplifying Expressions

Simplifying expressions is about making them as clean and straightforward as possible. It's akin to the goal of cleaning a messy room until it's neat and tidy. For mathematical expressions, simplifying means reducing expressions to their simplest form, where no further reduction is possible.
Here are some general steps to do this:

• Apply any known mathematical rules and laws, such as the negative exponent rule.
• Combine like terms where possible – terms that have the same base and exponent.
• Reduce fractions by dividing the numerator and the denominator by their greatest common divisor.

In our exercise, we took \( \frac{1}{a^{-2/3}} \) and used the rule to get \( a^{2/3} \), a much simpler form. Our expression started with an unnecessary fraction, and by applying the rules, we eliminated complexities.

Algebraic Fractions

Algebraic fractions appear regularly in algebra, just like regular fractions. They involve variables and can initially look intimidating. Understanding how to manipulate and simplify them is a valuable skill in algebra.
Some key points to remember about algebraic fractions include:

• They are expressed in the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials.
• Find the greatest common factor (GCF) to simplify the numerator and the denominator if possible.
• When variables have exponents, apply exponent rules to remove any negative exponents, as shown in our exercise.

For instance, starting with \( \frac{1}{a^{-2/3}} \), we observed that the denominator had a negative exponent. By using the negative exponent rule, we simplified it to \( a^{2/3} \), completely removing the fraction and resulting in a more straightforward expression.