Problem 81 - 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. $$ \left(\frac{2 a^{3}}{6 b}\right)^{4} $$

Step-by-Step Solution

Answer

\( \frac{a^{12}}{81b^4} \)

Step 1: Apply the Power of a Quotient Rule

The rule states that when raising a fraction to a power, both the numerator and the denominator should be raised to that power. This gives us \( \left(\frac{2a^3}{6b}\right)^4 = \frac{(2a^3)^4}{(6b)^4} \).

Step 2: Distribute the Power Over the Numerator

Apply the power of a product rule by raising both 2 and \( a^3 \) to the fourth power: \( (2a^3)^4 = 2^4 \cdot (a^3)^4 = 16a^{12} \).

Step 3: Distribute the Power Over the Denominator

Similarly, raise both 6 and \( b \) to the fourth power: \( (6b)^4 = 6^4 \cdot b^4 = 1296b^4 \).

Step 4: Write the Expression Using Positive Exponents

Combine the results from Steps 2 and 3 to write the expression as a single fraction: \( \frac{16a^{12}}{1296b^4} \).

Step 5: Simplify the Fraction

Simplify the fraction by dividing the numerator and the denominator by their greatest common factor, which is 16: \( \frac{16a^{12}}{1296b^4} = \frac{a^{12}}{81b^4} \).

Key Concepts

Power of a Quotient

The "Power of a Quotient" rule is a handy concept when you need to simplify expressions where a fraction is raised to a power. This rule tells us that each part of the fraction should be raised to the power separately. For example, if you have a fraction like \( \left(\frac{x}{y}\right)^n \), you would apply the power \( n \) to both the numerator \( x \) and the denominator \( y \).
Here’s how it works in practice:

• Imagine you have \( \left(\frac{2a^3}{6b}\right)^4 \).
• Apply the rule: raise both the numerator \( 2a^3 \) and the denominator \( 6b \) to the power of 4.

This allows you to rewrite the expression as \( \frac{(2a^3)^4}{(6b)^4} \), simplifying your work in the next steps.

Power of a Product

Using the "Power of a Product" rule helps when a product within parentheses is raised to a power. This means you will distribute the exponent across all factors inside the parentheses.
When you have \( (xy)^n \), you raise each factor to the power separately, resulting in \( x^n \times y^n \).
Applying this to the earlier example:

• The expression \( (2a^3)^4 \) becomes \( 2^4 \times (a^3)^4 \).
• This simplifies to \( 16 \times a^{12} \).

Using this rule makes it manageable to handle powers of terms, ensuring a correct and simplified result.

Fraction Simplification

Simplifying fractions is key to making mathematical expressions more manageable. The aim is to express the fraction with no common factors other than 1 in the numerator and denominator.
To simplify a fraction, follow these steps:

• Break down both the numerator and denominator into their prime factors if needed.
• Identify the greatest common factor, which is a shared factor in both.
• Divide both the numerator and the denominator by this greatest common factor.

In our example, the fraction \( \frac{16a^{12}}{1296b^4} \) simplifies down by dividing by their greatest common factor, which is 16, resulting in \( \frac{a^{12}}{81b^4} \). This efficiently reduces the expression to its simplest form.

Greatest Common Factor

The "Greatest Common Factor" (GCF) is crucial for fraction simplification, helping you reduce expressions to their simplest form. The GCF is the largest number that divides both the numerator and denominator without leaving a remainder.
Here's how you can find it:

• List out the factors of each number in the numerator and the denominator.
• Identify the largest factor that both numbers have in common.
• In our previous example, the GCF of 16 and 1296 is 16.

Understanding the GCF ensures that you are simplifying fractions correctly and efficiently, transforming expressions into cleaner, more efficient versions. This method is a powerful tool in any mathematician's toolkit.