Problem 82 - 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. $$ \left(\frac{-3 a^{2}}{9 b^{3}}\right)^{4} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{a^{8}}{81 b^{12}} \).

Step 1: Understand the Problem

We are given the expression \( \left( \frac{-3 a^{2}}{9 b^{3}} \right)^{4} \). We need to simplify this expression using the rules of exponents and express it using positive exponents.

Step 2: Apply Power of a Quotient Rule

The Power of a Quotient Rule states that \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \). Apply this rule to the expression:\[\left( \frac{-3 a^{2}}{9 b^{3}} \right)^{4} = \frac{(-3)^{4} (a^{2})^{4}}{(9)^{4} (b^{3})^{4}}\]

Step 3: Simplify the Numerator

Calculate \((-3)^4\) and \((a^2)^4\):- \((-3)^4 = 81\) because multiplying \(-3\) four times gives 81 (since \((-3) \times (-3) \times (-3) \times (-3) = 81\)).- \((a^2)^4 = a^{8}\) using the Power of a Power Rule \((a^m)^n = a^{m \cdot n}\).

Step 4: Simplify the Denominator

Calculate \(9^4\) and \((b^3)^4\):- \(9^4 = 6561\) because multiplying 9 four times gives 6561.- \((b^3)^4 = b^{12}\) using the Power of a Power Rule \((b^m)^n = b^{m \cdot n}\).

Step 5: Write the Simplified Expression

Now write the expression with the simplified numerator and denominator:\[\frac{81 a^{8}}{6561 b^{12}}\]

Step 6: Simplify the Fraction

Notice that 81 and 6561 can be further simplified. Divide both by 81:\[\frac{81}{6561} = \frac{1}{81}\]The final expression is:\[\frac{a^{8}}{81 b^{12}}\]

Key Concepts

Power of a Quotient Rule

The Power of a Quotient Rule is an essential tool in simplifying expressions with exponents. This rule shows how to handle expressions where a fraction is raised to a power. The rule states that when you have an expression of the form \( \left( \frac{a}{b} \right)^n \), it can be rewritten as \( \frac{a^n}{b^n} \). This means each part of the fraction is raised to the given power individually. This way, you distribute the power to both the numerator and the denominator separately.

Let's take our exercise as an example:

• The fraction \( \frac{-3 a^{2}}{9 b^{3}} \) is raised to the 4th power.
• According to the rule, apply the power of 4 to each component: \( \left( \frac{-3 a^{2}}{9 b^{3}} \right)^{4} = \frac{(-3)^4 (a^{2})^4}{(9)^4 (b^{3})^4} \).

This helps to break down complicated expressions into simpler parts, making it easier to further simplify the result. Always remember to distribute the power throughout all elements in the fraction.

Power of a Power Rule

The Power of a Power Rule is another fundamental principle in the realm of exponents. This rule tells us how to manage an exponent raised to another exponent. Specifically, for any base \(a\) with an exponent \(m\), if the whole expression is then raised to another exponent \(n\), you multiply the exponents: \((a^m)^n = a^{m \cdot n}\).

In our original exercise, the rule was crucial in simplifying both the numerator and the denominator:

• For the expression \((a^2)^4\), applying the rule gives us \(a^{2 \times 4} = a^8\).
• The same applies to \((b^3)^4\), resulting in \(b^{3 \times 4} = b^{12}\).

This technique is incredibly helpful in reducing seemingly complex powers down into a single, more manageable exponent. It's a straightforward multiplication of exponents that makes simplification easier.

Simplifying Expressions

Simplifying expressions is a crucial step in making mathematical problems more understandable and solvable. After applying the rules of exponents, you often end up with a result that can be further reduced to its simplest form.

For the final simplification in our exercise:

• The numerator \(81\) and denominator \(6561\) were simplified to \(\frac{1}{81}\) by dividing both by 81.
• Combining this with the simplified powers, the expression becomes \(\frac{a^8}{81 b^{12}}\).

By reducing fractions and eliminating common factors, you reach an expression that is concise and easier to work with. Simplifying ensures that your final answer is in the simplest, most efficient form, using only positive exponents. This not only aids in understanding but also ensures clarity and precision in communication.