Problem 23 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Simplify the expression, writing your answer using positive exponents only. $$ \frac{3 x^{3}}{2 x^{4}} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{3}{2x}\).

Step 1: Recall the rules of exponents

When dividing terms with the same base, subtract the exponents. In this case, the base is x. So, we will apply this rule to simplify the expression.

Step 2: Divide the terms

We will divide the coefficients (the numbers) and the terms with the same base (the x's). This operation will result in: \[ \frac{3}{2} \cdot \frac{x^3}{x^4} \]

Step 3: Subtract the exponents

The next step is to subtract the exponents of the terms with the same base, x. \[ \frac{3}{2} \cdot x^{3-4} \]

Step 4: Simplify the expression

Now, we simplify the expression by calculating the exponent. \[ \frac{3}{2} \cdot x^{-1} \]

Step 5: Rewrite using positive exponents only

To rewrite the expression using positive exponents only, we move the term with the negative exponent to the denominator. \[ \frac{3}{2x} \] The final simplified expression is \(\frac{3}{2x}\).

Key Concepts

Exponent Rules

Exponent rules are fundamental for simplifying expressions with powers. A key rule is when dividing like terms with the same base, subtract their exponents. For example, consider \(\frac{x^a}{x^b} = x^{a-b}\). This rule simplifies expressions by reducing the power of the variable. Remember that all terms must have the same base to apply this rule effectively.
Subtraction of exponents is just the beginning. It helps in understanding more complex expressions where multiple exponents interact. Pay attention to the base of each term as this defines how and when you can apply these rules.

Positive Exponents

Expressions are often required to be simplified using only positive exponents. Negative exponents indicate reciprocal or division.
For instance, \(x^{-n}\) is equivalent to \(\frac{1}{x^n}\). This transformation eliminates negative exponents and leaves the expression cleaner and sometimes simpler to use.
Writing with positive exponents is essential for clarity. In the given exercise, rewriting \(x^{-1}\) as \(\frac{1}{x}\) allows for a straightforward interpretation, easing evaluation and further calculations.

Division of Algebraic Expressions

Dividing algebraic expressions involves breaking down both coefficients and variable terms. Begin by dividing the coefficients: the numbers in front of variables. For the expression \(\frac{3}{2}\), it's straightforward.
Next, tackle the variable terms following exponent rules. With the same base, subtract the exponents to simplify these terms. In this example \(x^{3-4}\), which becomes \(x^{-1}\).
Finally, ensure the expression is written properly. Simplifying includes rewriting any negative exponents using positive terms. This often involves placement in the denominator, turning \(x^{-1}\) into \(\frac{1}{x}\), leaving the final result as \(\frac{3}{2x}\).
Division might initially seem challenging, but breaking it into manageable steps ensures accuracy, achieving a clean, simplified form.