Problem 75 - College Algebra with Modeling and Visualization

Question

Exercises \(65-90:\) Use rules of exponents to simplify the expression. Use positive exponents to write your answer. $$ \frac{\left(d^{3}\right)^{-2}}{\left(d^{-2}\right)^{3}} $$

Step-by-Step Solution

Answer

The simplified expression is 1.

Step 1: Apply the Power of a Power Rule

The expression is \( \frac{\left(d^{3}\right)^{-2}}{\left(d^{-2}\right)^{3}} \). First, apply the power of a power rule, which states \((a^m)^n = a^{m \cdot n}\). For the numerator, apply this rule: \((d^3)^{-2} = d^{3 \cdot (-2)} = d^{-6}\). For the denominator: \((d^{-2})^3 = d^{-2 \cdot 3} = d^{-6}\).

Step 2: Simplify the Fraction

Now substitute the simplified terms back into the expression: \( \frac{d^{-6}}{d^{-6}} \). Since both the numerator and the denominator are the same, the entire fraction simplifies to 1. By the rule \( \frac{a^m}{a^m} = a^{m-m} = a^0 = 1\) for any non-zero \(a\), the expression is 1.

Key Concepts

Power of a Power Rule

In mathematics, the Power of a Power Rule is a crucial part of understanding how to manipulate expressions with exponents. When you have a base raised to an exponent, and then that entire expression is raised to another power, you must multiply the exponents. The general formula is \[(a^m)^n = a^{m imes n}\].
For example, let's consider the expression \((d^3)^{-2}\) as in our exercise. Applying the Power of a Power Rule involves multiplying the two exponents: \(3\) (from \(d^3\)) and \(-2\) (the outer exponent). Thus, \((d^3)^{-2} = d^{3 imes (-2)} = d^{-6}\).

• Use multiplication on exponents when dealing with \((a^m)^n\).
• Apply this rule separately to the numerator and the denominator in expressions.

This rule simplifies expressions by reducing complex stacked exponents into simpler, single expressions.

Simplifying Expressions

Simplifying expressions in algebra often involves reducing complex expressions into their simplest form. For fraction expressions involving exponents, this can mean canceling out similar terms. In our example with the expression \(\frac{d^{-6}}{d^{-6}}\), we notice that the base and exponent in both the numerator and the denominator are the same.
This means that the entire fraction can be simplified to 1 because of the rule: if you divide a term by itself, it equals 1. Therefore, \(\frac{d^{-6}}{d^{-6}} = d^{-6 - (-6)} = d^0 = 1\).

• Simplify fractions by canceling out identical bases in the numerator and the denominator.
• Remember, anything raised to the power of \(0\) results in \(1\).

This is a fundamental part of simplifying expressions: recognizing and applying rules efficiently.

Positive Exponents

Positive exponents are particularly useful and preferred in mathematical expressions due to their straightforward nature and clarity. In cases where expressions result in negative exponents, we typically convert them to positive exponents for a final answer. This is also required for writing final answers, keeping expressions clean and standard.
Here's a quick look at how to convert: if you have a negative exponent \(a^{-m}\), you can rewrite it as \(\frac{1}{a^m}\). However, in a simplified expression like our example, converting it to 1 effectively deals with any remaining negative exponents.

• Negative exponents indicate reciprocation of the base term.
• Convert negative exponents to positive by taking the reciprocal.

Ultimately, working towards positive exponents ensures that expressions remain intuitive and accessible.