Problem 24 - Basic Engineering Mathematics

Question

Simplify \(\frac{d^{2} e^{2} f^{1 / 2}}{\left(d^{3 / 2} e f^{5 / 2}\right)^{2}}\) expressing the answer with positive indices only.

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{d f^{9/2}}\).

Step 1: Simplify the Denominator

First, simplify the expression within the parentheses in the denominator. The denominator is \((d^{3/2} e f^{5/2})^2\). Use the power of a product rule: \[(a b c)^n = a^n b^n c^n\]Thus, \[(d^{3/2} e f^{5/2})^2 = (d^{3/2})^2 (e)^2 (f^{5/2})^2\]Simplify each term:\[(d^{3/2})^2 = d^{3}, \quad (e)^2 = e^2, \quad (f^{5/2})^2 = f^5\] So the denominator becomes \(d^3 e^2 f^5\).

Step 2: Setup the Fraction with Simplified Denominator

Replace the original denominator with the simplified one in the fraction:\[\frac{d^{2} e^{2} f^{1/2}}{d^3 e^2 f^5}\]

Step 3: Simplify the Fraction

Divide each term in the numerator by the corresponding term in the denominator. Use the quotient of powers rule:\[a^m/a^n = a^{m-n}\]Apply this for each variable:- \(d^{2}/d^3 = d^{2-3} = d^{-1}\)- \(e^{2}/e^2 = e^{2-2} = e^{0} = 1\) (since any number raised to the 0 power is 1)- \(f^{1/2}/f^5 = f^{1/2-5} = f^{-9/2}\) Now we combine these simplified results into:\[d^{-1} \cdot 1 \cdot f^{-9/2} = d^{-1} f^{-9/2}\]

Step 4: Express with Positive Indices

To express with positive indices, apply the rule that \(a^{-n} = \frac{1}{a^n}\):- Convert \(d^{-1}\) to \(\frac{1}{d^1} = \frac{1}{d}\)- Convert \(f^{-9/2}\) to \(\frac{1}{f^{9/2}}\)Combine these into a single expression:\[\frac{1}{d f^{9/2}}\]

Key Concepts

Exponent Rules

Exponent rules are essential tools that make it easier to work with powers of numbers or variables. Here are some key rules you should know:

• Product of Powers Rule: When multiplying two expressions with the same base, sum the exponents. For example, if you have \(a^m \times a^n\), it simplifies to \(a^{m+n}\).
• Power of a Power Rule: To raise a power to another power, multiply the exponents. For instance, \((a^m)^n = a^{m \cdot n}\).
• Power of a Product Rule: When a product is raised to an exponent, each factor in the product is raised to that exponent separately. For example, \((abc)^n = a^n b^n c^n\).

Using these rules, we can handle complex algebraic expressions and simplify them to make calculations easier. We applied these rules in the given problem to simplify the denominator before proceeding with further steps.

Simplification of Expressions

Simplification of algebraic expressions is about reducing an expression to its simplest form while keeping its value unchanged. This involves combining like terms and canceling common factors. In the context of fractions with algebraic expressions, the numerator and the denominator should be simplified separately using exponent rules.

To simplify expressions effectively, follow these general steps:

• Simplify powers in parentheses first by applying the Power of a Product Rule.
• Apply any relevant exponent rules to simplify the expression further.
• Break down the expression into manageable parts and look for common factors to cancel.

For example, in our given expression, we simplified the denominator by distributing the power over each term individually, making the expression more manageable to handle. This allows easy application of the quotient of powers rule later on.

Quotient of Powers Rule

The quotient of powers rule is an exponent rule used when dividing two expressions with the same base. The rule states that you subtract the exponent in the denominator from the exponent in the numerator: \[a^m / a^n = a^{m-n}\]

This rule helps simplify expressions by reducing powers and making them easier to calculate.
To utilize this rule effectively, follow these tips:

• Ensure that the bases in the numerator and denominator match before applying the rule.
• Perform the subtraction carefully to avoid mistakes, especially with negative results.
• Remember that any expression raised to the zero power equals one, which often simplifies parts of the expression considerably.

Applying the quotient of powers rule is crucial when simplifying fractions and achieving expressions in terms of positive indices. In the problem, we used this rule to simplify the given fraction, turning it from a complex expression into an easily readable format.