Problem 67 - Algebra and Trigonometry

Question

\(47-72\) . Simplify the expression, and eliminate any negative exponent(s). $$ \left(\frac{3 a}{b^{3}}\right)^{-1} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{b^3}{3a} \).

Step 1: Apply Negative Exponent Rule

The expression given is \( \left(\frac{3a}{b^3}\right)^{-1} \). When you have a negative exponent, you can invert the fraction to make the exponent positive. \( \left(\frac{3a}{b^3}\right)^{-1} = \left(\frac{b^3}{3a}\right)\) with an exponent of 1.

Step 2: Simplify the Expression

Since the new exponent is 1, the expression simplifies to \( \frac{b^3}{3a} \). There are no common factors in the numerator and denominator to simplify further.

Key Concepts

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its simplest form. This means that the numerator (top number) and the denominator (bottom number) of the fraction cannot be divided by any common factor other than 1. Simplifying a fraction makes it easier to work with and understand. For example, if you have a fraction like \( \frac{6}{8} \), you can divide both the numerator and the denominator by their greatest common factor, which is 2 in this case. This simplifies the fraction to \( \frac{3}{4} \).
Simplifying fractions is essential in algebra and higher mathematics because it keeps equations tidy and helps you see the relationships between numbers more clearly. When working with algebraic fractions, like in our exercise, you will also look for common algebraic factors to simplify whenever possible.

Negative Exponent Rule

The negative exponent rule is a fundamental concept in algebra that helps us understand how to handle terms with negative exponents. According to the rule, a negative exponent indicates that the base should be inverted. Concretely, any number or expression with a negative exponent can be rewritten by flipping it to its reciprocal with a positive exponent.

• For any nonzero number \( x \) with exponent \( -n \), it is equivalent to \( \frac{1}{x^n} \).
• Thus, \( x^{-3} = \frac{1}{x^3} \).

In our given exercise, the expression \( \left(\frac{3a}{b^3}\right)^{-1} \) is tackled by applying this rule. The entire fraction is inverted so that the negative exponent becomes positive, making it \( \left(\frac{b^3}{3a}\right)^1 \). This conversion simplifies further calculations and helps eliminate negative exponents from expressions, which is often desirable for simplifying mathematical operations.

Invert Fraction

Inverting a fraction means to "flip" the fraction, swapping the positions of the numerator and the denominator. This action is especially useful when dealing with negative exponents, as it makes the exponent positive. In the context of our exercise, inverting the fraction \( \frac{3a}{b^3} \) changed it to \( \frac{b^3}{3a} \).
This step is vital because it simplifies handling negative exponents in algebraic operations, as discussed under the negative exponent rule.
Understanding how to invert fractions and why it's important helps simplify mathematical expressions and solve problems more efficiently. Inverting is not just limited to negative exponents; it also plays a role in fraction division, helping to solve expressions by turning division into multiplication by a reciprocal.