Problem 42 - Algebra and Trigonometry

Question

Simplify the expression and eliminate any negative exponent(s). $$ \left(2 u^{2} v^{3}\right)^{3}\left(3 u^{3} v\right)^{-2} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{8}{9} v^{7} \).

Step 1: Apply Exponent Rule for Powers

Use the exponent rule \( (a^m)^n = a^{m\cdot n} \) to simplify each part of the expression separately. For the first term, \( \left(2 u^{2} v^{3}\right)^{3} \), distribute the exponent 3 to each factor: \( 2^{3} (u^{2})^{3} (v^{3})^{3} \). For the second term, \( \left(3 u^{3} v\right)^{-2} \), distribute the exponent \(-2\) to each factor: \( 3^{-2} (u^{3})^{-2} (v)^{-2} \).

Step 2: Simplify Each Term with the Power Applied

Calculate the powers obtained from Step 1. For the first term: \( 2^{3} = 8 \), \( (u^{2})^{3} = u^{6} \), and \( (v^{3})^{3} = v^{9} \). Thus, the expression becomes \( 8 u^{6} v^{9} \). For the second term: \( 3^{-2} = \frac{1}{9} \), \( (u^{3})^{-2} = u^{-6} \), and \( (v)^{-2} = v^{-2} \). Thus, it becomes \( \frac{1}{9} u^{-6} v^{-2} \).

Step 3: Multiply the Simplified Terms

To combine \(8 u^{6} v^{9}\) and \(\frac{1}{9} u^{-6} v^{-2}\), multiply the coefficients and add the exponents for like bases. The coefficient is \(8 \cdot \frac{1}{9} = \frac{8}{9}\). For the \(u\) terms, apply \((u^{6})(u^{-6}) = u^{0} = 1\). For the \(v\) terms, \(v^{9}\cdot v^{-2} = v^{7}\).

Step 4: Final Simplification

Combine the results of the previous steps to obtain the final simplified expression. The expression becomes: \(rac{8}{9} v^{7} \).

Key Concepts

Exponent Rules

Exponent rules are essential in simplifying expressions that involve powers, making calculations more manageable. Here’s a breakdown of some critical rules:

• Product of Powers Rule: When multiplying two powers with the same base, you add the exponents: \(a^m \times a^n = a^{m+n}\).
• Power of a Power Rule: When taking a power to another power, you multiply the exponents: \((a^m)^n = a^{m \cdot n}\). This helps in simplifying expressions like \((u^2)^3\), giving us \(u^6\).
• Power of a Product Rule: When you have a product raised to a power, distribute the power to each factor: \((ab)^m = a^m b^m\).

Applying these rules can transform a complex expression into something much easier to work with. In our original exercise, the exponent rules were used to distribute the powers across different terms, simplifying the multiplications involved with each base.

Negative Exponents

Negative exponents can seem a bit tricky at first, but they follow a straightforward pattern. A negative exponent indicates that the base is on the wrong side of the fraction line, and needs to be flipped.

• Basic Rule: \(a^{-n} = \frac{1}{a^n}\). This represents the reciprocal. So, \(3^{-2} = \frac{1}{3^2} = \frac{1}{9}\).
• In Expressions: Negative exponent rules are applied similarly when simplifying terms like \((u^3)^{-2}\). This converts to \(u^{-6}\), which is equal to \(\frac{1}{u^6}\).

These rules are crucial for eliminating negative exponents and expressing everything with positive exponents, which often makes further calculations more straightforward, as illustrated in the solution of our exercise.

Distributive Property

The Distributive Property is a valuable tool in algebra for expanding expressions where a term outside the parenthesis multiplies each term inside. It can be defined simply as:

• General Form: \(a(b + c) = ab + ac\).
• Involving Exponents: When dealing with expressions like \((2u^2v^3)^3\), the property helps in distributing the power 3 to each factor, becoming \(2^3 \cdot (u^2)^3 \cdot (v^3)^3 \).

Using the Distributive Property effectively allows for breaking down complex terms into manageable parts, immediately simplifying the overall expression. In the given exercise, applying this property helped distribute both positive and negative exponents, ensuring every base was treated separately, ultimately leading to a simplified form.