Problem 44 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-46: Simplify. $$ \left(\frac{c^{-4}}{16 d^{8}}\right)^{3 / 4} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{1}{8 c^3 d^6} \).

Step 1: Apply the Power of a Quotient Rule

The expression given is \( \left(\frac{c^{-4}}{16 d^{8}}\right)^{3/4} \). The Power of a Quotient Rule states \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \). Applying this rule, we get: \[ \frac{(c^{-4})^{3/4}}{(16 d^{8})^{3/4}} \]

Step 2: Apply the Power of a Power Rule

For both numerator and denominator, use the Power of a Power Rule: \( (a^m)^n = a^{m \cdot n} \). In the numerator: \((c^{-4})^{3/4} = c^{-4 \cdot \frac{3}{4}} = c^{-3}\). In the denominator: \((16 d^{8})^{3/4} = 16^{3/4} \cdot (d^{8})^{3/4} = 16^{3/4} \cdot d^{6} \).

Step 3: Simplify the Numerical Expression

Evaluate \(16^{3/4}\). Recall that \(16^{3/4} = (16^{1/4})^3\). Since \(16^{1/4} = 2\) (because \(2^4 = 16\)), \(16^{3/4} = 2^3 = 8\). So their expression becomes:\[ \frac{c^{-3}}{8 d^{6}} \]

Step 4: Simplify Further Using Positive Exponents

Rewrite \(c^{-3}\) with positive exponents, using \(a^{-m} = \frac{1}{a^m}\):\[ \frac{1}{c^3} \times \frac{1}{8 d^6} = \frac{1}{8 c^3 d^6} \]

Step 5: Conclusion: Final Simplified Expression

The fully simplified expression is:\[ \frac{1}{8 c^3 d^6} \] This is the simplest form of the original problem.

Key Concepts

Power of a Quotient Rule

When dealing with exponents, the Power of a Quotient Rule is a handy tool. This rule is used to simplify expressions where a fraction is raised to a power. It states that:

• \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \).

This means you can separately apply the exponent to both the numerator and the denominator.
Imagine you have a pie cut into slices. If each slice (numerator) gets x toppings, and the entire pie (denominator) is multiplied by that same factor, you are essentially distributing the power to each part of the fraction.
Let's see this concept in action: using \( \left(\frac{c^{-4}}{16 d^{8}}\right)^{3/4} \), the Power of a Quotient Rule allows us to distribute \( \frac{3}{4} \) as follows:

• Numerator: \( (c^{-4})^{3/4} \)
• Denominator: \( (16 d^{8})^{3/4} \)

This step is crucial as it paves the way for further simplification.

Power of a Power Rule

The Power of a Power Rule helps simplify expressions where an exponent is raised to another power. It states:

• \( (a^m)^n = a^{m \cdot n} \).

The way it works is simple: multiply the exponents. Think of it as stacking powers. If you have a stack of blocks and double it, you've effectively multiplied the number of blocks.
In our exercise:

• For \((c^{-4})^{3/4}\), it becomes \(c^{-4 \cdot \frac{3}{4}} = c^{-3}\).
• For \((16 d^{8})^{3/4}\), distribute the fraction, leading to \(16^{3/4} \cdot d^{6}\).

The Power of a Power Rule simplifies these expressions, making them manageable for further steps.

Negative Exponents

Negative exponents can initially seem confusing, but they actually simplify expressions by indicating reciprocals. The rule is straightforward:

• \(a^{-m} = \frac{1}{a^m}\).

This means that instead of having a negative exponent, you can rewrite it as a positive exponent in a fraction's denominator.
In our example, \(c^{-3}\) transforms to \(\frac{1}{c^3}\). This systematically shifts the expression from negative to positive, which is often simpler and clearer to work with in equations.
Understanding negative exponents allows us to express the final answer in a clean, positive form, reducing the complexity of the expression further.

Simplifying Expressions

Simplifying expressions is key to reducing complex mathematical problems into their most understandable form. When simplifying, you look to combine like terms and reduce them to their simplest terms. This process involves rules such as those for exponents, and often results in a tidier form.
Our exercise walked through simplification using the determined components:

• The Power of a Quotient Rule broke down the fraction.
• The Power of a Power Rule adjusted the terms to simpler exponents.
• Negative exponents were converted into positive ones using fractions.

Finally, all these steps led to calculating the numerical value, such as \(16^{3/4} = 8\).
This turns our expression from \(\frac{c^{-3}}{8 d^{6}}\) to a fully simplified form: \(\frac{1}{8 c^3 d^6}\).
Simplification makes expressions easier to handle and understand, crucial for accurately solving mathematical problems.