Problem 41 - A Graphical Approach to College Algebra

Question

Perform the indicated operations. Write your answers with only positive exponents. Assume that all variables represent positive real numbers. $$\left(\frac{-a}{b^{-3}}\right)^{-1}$$

Step-by-Step Solution

Answer

\( \frac{1}{-a \cdot b^3} \)

Step 1: Simplify the Fraction

Start with the expression \( \left(\frac{-a}{b^{-3}}\right)^{-1} \). The fraction inside the parentheses can be rewritten by moving \( b^{-3} \) to the numerator, giving us \( \left(-a \cdot b^3\right)^{-1} \).

Step 2: Apply the Negative Exponent

Now, apply the negative exponent \(-1\) to the expression \(-a \cdot b^3\). The rule is that \( (x)^{-n} = \frac{1}{x^n} \). Therefore, we get \( \frac{1}{-a \cdot b^3} \).

Step 3: Convert to Positive Exponents

We already have the expression \( \frac{1}{-a \cdot b^3} \), and since \( -a \cdot b^3 \) has only positive exponents, we're done.

Key Concepts

Negative Exponents

A negative exponent means that the base of that exponent is on the "wrong side" of the fraction line. When we see a negative exponent, it indicates that the reciprocal of the base should be taken. For example, for a term like \((x)^{-n}\), we would convert this into \(\frac{1}{x^n}\). In simpler terms, a base with a negative exponent is moved from the numerator to the denominator or vice versa, and the sign of the exponent becomes positive.

In the given problem \(\left(\frac{-a}{b^{-3}}\right)^{-1}\), the \(b^{-3}\) is in the denominator, and to make the exponent positive, it gets moved to the numerator resulting in \(-a \cdot b^3\). The overall negative exponent \(-1\) applies outside the parentheses, flipping the expression inside. Thus, it ends up in the denominator as \(\frac{1}{-a \cdot b^3}\). Being comfortable with reciprocal operations is the key to mastering negative exponents.

Positive Exponents

A positive exponent simply tells you how many times to multiply the base by itself. For instance, \(x^n\) implies multiplying \(x\) by itself \(n\) times. Positive exponents are straightforward because they involve repeated multiplication without the need to flip anything.

Using positive exponents can simplify expressions and make equations easier to work with. In solving \(\left(\frac{-a}{b^{-3}}\right)^{-1}\), once we have moved the \(b^{-3}\) to the numerator as \(b^3\), it becomes a simple multiplication, \(-a \cdot b^3\). The result with positive exponents is easy to read and work with further. We don't need to do anything else with plus signs, unlike negative exponents, which require careful handling to avoid mistakes.

Simplifying Expressions

Simplifying algebraic expressions involves reducing them to their simplest form by applying algebraic rules and properties. This often includes combining like terms, reducing fractions, and managing exponents. The goal is clarity and efficiency in expression.

In the provided scenario, the aim was to write the expression \(\left(\frac{-a}{b^{-3}}\right)^{-1}\) with only positive exponents. First, by understanding and using the property of negative exponents, we rearranged \(b^{-3}\) from the denominator to the numerator as \(b^3\), which simplified the expression inside to \(-a \cdot b^3\). Applying another negative exponent outside flipped everything inside, resulting in \(\frac{1}{-a \cdot b^3}\).

A simplified expression is always preferred in mathematics because it provides a cleaner, more readable, and more easily operable format. This enables more efficient problem-solving and clearer communication.