Problem 44 - Precalculus: Functions and Graphs

Question

Simplify. $$\left(\frac{c^{-4}}{16 d^{3}}\right)^{3 / 4}$$

Step-by-Step Solution

Answer

\(\frac{1}{8c^3d^{9/4}}\)

Step 1: Apply the Power of a Quotient Property

The expression given is \(\left(\frac{c^{-4}}{16 d^{3}}\right)^{3 / 4}\). First, apply the property \((\frac{a}{b})^n = \frac{a^n}{b^n}\). Thus, the expression becomes \(\frac{(c^{-4})^{3/4}}{(16d^3)^{3/4}}\).

Step 2: Simplify the Numerator Using Power of a Power Property

For the numerator \((c^{-4})^{3/4}\), use the power of a power property \((a^m)^n = a^{m\cdot n}\). So, \((c^{-4})^{3/4} = c^{-4 \cdot \frac{3}{4}} = c^{-3}\).

Step 3: Simplify the Denominator Using Power of a Power Property

For the denominator \((16d^3)^{3/4}\), separate the terms into \(16^{3/4}\) and \((d^3)^{3/4}\). Simplify each term individually: \(16^{3/4} = (2^4)^{3/4} = 2^{4\cdot \frac{3}{4}} = 2^3 = 8\) and \((d^3)^{3/4} = d^{3 \cdot \frac{3}{4}} = d^{9/4}\).

Step 4: Combine Simplified Numerator and Denominator

After simplifying, the numerator \(c^{-3}\) remains unchanged, and the denominator becomes \(8d^{9/4}\). Assemble them back into a fraction: \(\frac{c^{-3}}{8d^{9/4}}\).

Step 5: Express Final Answer

Since \(c^{-3} = \frac{1}{c^3}\), rewrite the expression: \(\frac{1}{c^3}\) over \(8d^{9/4}\). Combine the fractions to obtain the simplified result: \(\frac{1}{8c^3d^{9/4}}\).

Key Concepts

Power of a Quotient Property

When simplifying expressions involving exponents, one powerful tool is the Power of a Quotient Property. This property is very helpful when you have an exponent outside of a fraction. The notation looks like this:

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

This means that you can apply the exponent to both the numerator and the denominator separately. This is especially useful because it allows us to handle complex expressions one part at a time.
For example, in the exercise \( \left(\frac{c^{-4}}{16 d^{3}}\right)^{3/4} \), we used this property to separate it into \( \frac{(c^{-4})^{3/4}}{(16d^3)^{3/4}} \). Now, each part can be worked on individually, making the simplifying process much easier.

Power of a Power Property

The Power of a Power Property is another essential rule for dealing with expressions that have exponents. If you see repeated exponents, like \((a^m)^n\), you can simplify it by multiplying the exponents:

• \( (a^m)^n = a^{m \cdot n} \).

This property speeds up the process of dealing with complex power expressions.
In the exercise, we apply this concept to both the numerator and the denominator. For the numerator, \((c^{-4})^{3/4}\), applying the property gives us \(c^{-4 \cdot \frac{3}{4}} = c^{-3}\). For the denominator \((d^3)^{3/4}\), it results in \(d^{9/4}\). Each step considerably simplifies the expression, and brings us closer to the final answer.

Fractional Exponents

Fractional exponents might seem tricky at first, but they are just another way to express roots and powers in mathematics. Understanding them is crucial for simplifying expressions.

• An exponent \( \frac{m}{n} \) is equivalent to the \(n\)th root of a quantity raised to the \(m\)th power: \( a^{\frac{m}{n}} = \sqrt[n]{a^m} \).

When you encounter a fraction in an exponent, you can think of it as doing two operations: taking a root and raising to a power. In the exercise, we see this when calculating \(16^{3/4}\), since \(16 = 2^4\). This simplifies to \((2^4)^{3/4} = 2^{4 \cdot \frac{3}{4}} = 2^3\), which equals 8. Isn't that neat? Breaking down fractional exponents is straightforward when you understand what each component of the fraction signifies.

Exponents

Exponents are fundamental in mathematics as they indicate how many times to use the number in a multiplication. They are expressed using a small number to the top right of the base number, such as \(a^n\), which means \(a\) multiplied by itself \(n\) times. Understanding basic exponent rules can make handling expressions a breeze:

• Product of Powers: \( a^m \cdot a^n = a^{m+n} \)
• Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \)
• Power of a Product: \( (ab)^n = a^n \cdot b^n \)

In our exercise, we utilized various properties of exponents to achieve simplification. For instance, by understanding \(c^{-3} = \frac{1}{c^3}\), we easily adjusted the fraction to fit the final answer. Remember, exponents help to quickly reduce large and complex multiplication problems into simpler forms.