Simplify ((x^2)/9+1/4)/(6x)

The given problem asks to perform an algebraic simplification of a complex rational expression. The expression consists of a numerator that includes two terms added together, one of which is a variable (x) squared and divided by 9, and the other is a constant (1) divided by 4. This entire sum in the numerator is then to be divided by the term '6x' which is in the denominator. The task is to apply the rules of simplification such as combining like terms, factoring, and reducing fractions to write the expression in its simplest form.

$\frac{\frac{x^{2}}{9} + \frac{1}{4}}{6 x}$

Answer

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Solution:

Step 1: Simplify the numerator.

Step 1.1: Convert $\frac{x^{2}}{9}$ to have the same denominator as $\frac{1}{4}$ by multiplying it by $\frac{4}{4}$ .

$\frac{\frac{x^{2}}{9} \cdot \frac{4}{4} + \frac{1}{4}}{6 x}$
Step 1.2: Similarly, convert $\frac{1}{4}$ to have the same denominator as $\frac{x^{2}}{9}$ by multiplying it by $\frac{9}{9}$ .

$\frac{\frac{x^{2}}{9} \cdot \frac{4}{4} + \frac{1}{4} \cdot \frac{9}{9}}{6 x}$
Step 1.3: Establish a common denominator of $36$ for both fractions in the numerator.
- Step 1.3.1: Multiply $\frac{x^{2}}{9}$ by $\frac{4}{4}$ .
  
  $\frac{\frac{x^{2} \cdot 4}{9 \cdot 4} + \frac{1}{4} \cdot \frac{9}{9}}{6 x}$
- Step 1.3.2: Calculate $9 \cdot 4$ .
  
  $\frac{\frac{x^{2} \cdot 4}{36} + \frac{1}{4} \cdot \frac{9}{9}}{6 x}$
- Step 1.3.3: Multiply $\frac{1}{4}$ by $\frac{9}{9}$ .
  
  $\frac{\frac{x^{2} \cdot 4}{36} + \frac{9}{4 \cdot 9}}{6 x}$
- Step 1.3.4: Calculate $4 \cdot 9$ .
  
  $\frac{\frac{x^{2} \cdot 4}{36} + \frac{9}{36}}{6 x}$
Step 1.4: Combine the terms in the numerator over the common denominator.

$\frac{\frac{4 x^{2} + 9}{36}}{6 x}$
Step 1.5: Rearrange the terms in the numerator.

$\frac{\frac{4 x^{2} + 9}{36}}{6 x}$

Step 2: Multiply the numerator by the reciprocal of the denominator.

$\frac{4 x^{2} + 9}{36} \cdot \frac{1}{6 x}$

Step 3: Simplify the expression.

Step 3.1: Multiply $\frac{4 x^{2} + 9}{36}$ by $\frac{1}{6 x}$ .

$\frac{4 x^{2} + 9}{36 \cdot 6 x}$
Step 3.2: Multiply $36$ by $6$ .

$\frac{4 x^{2} + 9}{216 x}$

Knowledge Notes:

To simplify a complex fraction, you can follow these steps:

Simplify the numerator and denominator separately if possible.
Find a common denominator for all fractions within the numerator and/or denominator.
Combine the fractions in the numerator and/or denominator using the common denominator.
Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator.
Simplify the resulting expression if possible.

In this problem, we worked with algebraic expressions involving fractions. We used the property that multiplying a fraction by another fraction that is equivalent to 1 (such as $\frac{4}{4}$ or $\frac{9}{9}$ ) does not change its value, but allows us to find a common denominator. We also used the distributive property to combine like terms in the numerator. Finally, we simplified the complex fraction by multiplying the numerator by the reciprocal of the denominator.