Problem 33 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Simplify the expression, writing your answer using positive exponents only. $$ \left(\frac{2 u^{2} v^{3}}{3 u v}\right)^{-1} $$

Step-by-Step Solution

Answer

The simplified expression, with no negative exponents, is \(\frac{3}{2 u v^{2}}\).

Step 1: Handle the negative exponent

Recall that a negative exponent means taking the reciprocal of the base. In this scenario, simply reverse the numerator and the denominator (flip the fraction) and change the negative exponent to a positive exponent: \[ \left(\frac{2 u^{2} v^{3}}{3 u v}\right)^{-1} = \left(\frac{3 u v}{2 u^{2} v^{3}}\right)^{1} \]

Step 2: Simplify the fraction

Now, we can simplify the fraction by cancelling out common factors: Since there's a factor of 'u', we can cancel it from both the numerator and the denominator: \[ \left(\frac{3 \cancel{u} v}{2 u^{\cancel{2-1}} v^{3}}\right) \] Similarly, we can cancel out a factor of 'v': \[ \left( \frac{3}{2 u v^{\cancel{3-1}}} \right) \] The simplified expression is: \[ \frac{3}{2 u v^{2}} \] That's the final answer - it has been simplified, and there are no negative exponents remaining.

Key Concepts

Exponents

Exponents are numbers that indicate how many times a number, known as the base, is multiplied by itself. In the given problem, exponents appear in the terms such as \( u^2 \) and \( v^3 \). When handling exponents, it's crucial to follow a few basic rules:

• A negative exponent means you should take the reciprocal of the base and make the exponent positive. For example, \( a^{-n} = \frac{1}{a^n} \).
• When multiplying terms with the same base, you add their exponents: \( a^m \times a^n = a^{m+n} \).
• When dividing terms with the same base, you subtract their exponents: \( \frac{a^m}{a^n} = a^{m-n} \).

Applying these rules helps transition from a negative exponent to a positive exponent, simplifying complex expressions step by step.

Simplifying Expressions

Simplifying expressions involves reducing a mathematical expression to its simplest form. This means performing all possible arithmetic and reducing fractions, all while following the order of operations.

• Start by resolving any operations within parentheses.
• Apply exponent rules, especially if there are any negative exponents involved.
• Find and cancel any common factors, just like we can see with the terms containing 'u' and 'v' in both the numerator and the denominator.

In the provided exercise, the expression \( \left(\frac{2 u^{2} v^{3}}{3 u v}\right)^{-1} \) was first flipped due to the negative exponent and then had its common factors canceled to reach the simple form \( \frac{3}{2 u v^{2}} \). This makes it easier to understand and work with further.

Fraction Operations

Working with fractions involves performing basic operations such as addition, subtraction, multiplication, and division. For the given exercise, it's important to understand:

• Flipping the fraction: This method is used when multiplying by a reciprocal, especially when working with negative exponents as seen in the solution \( \left(\frac{2 u^{2} v^{3}}{3 u v}\right)^{-1} = \left(\frac{3 u v}{2 u^{2} v^{3}}\right)^{1} \).
• Factor cancellation: Find common factors in the numerator and denominator to simplify the fraction further, such as canceling 'u' and 'v'.

Remember, the goal of fraction operations, particularly in expressions involving variables, is to simplify wherever possible, ensuring the expression is concise and as reduced as feasible.