Problem 54 - Intermediate Algebra

Question

Simplify each of the numerical expressions. $$ 4\left(\frac{1}{3}\right)^{3}+3\left(\frac{1}{3}\right)^{2}+2\left(\frac{1}{3}\right)+6 $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{193}{27} \).

Step 1: Simplify the Power of Fractions

First, calculate each power of the fraction \( \left( \frac{1}{3} \right) \). For \( \left( \frac{1}{3} \right)^3 \), multiply three times: \( \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27} \). For \( \left( \frac{1}{3} \right)^2 \), multiply twice: \( \frac{1}{3} \times \frac{1}{3} = \frac{1}{9} \).

Step 2: Apply Multiplication to Each Term

Substitute the calculated fractions back into the original expression: \( 4 \times \frac{1}{27} + 3 \times \frac{1}{9} + 2 \times \frac{1}{3} + 6 \).

Step 3: Perform Each Multiplication

Calculate each multiplication: \( 4 \times \frac{1}{27} = \frac{4}{27} \), \( 3 \times \frac{1}{9} = \frac{3}{9} = \frac{1}{3} \), \( 2 \times \frac{1}{3} = \frac{2}{3} \).

Step 4: Find a Common Denominator and Add

Convert each fraction to have a common denominator of 27: \( \frac{1}{3} = \frac{9}{27} \) and \( \frac{2}{3} = \frac{18}{27} \). The expression becomes \( \frac{4}{27} + \frac{9}{27} + \frac{18}{27} + 6 \).

Step 5: Simplify the Fraction Addition

Add the fractions: \( \frac{4}{27} + \frac{9}{27} + \frac{18}{27} = \frac{31}{27} \).

Step 6: Add the Whole Number

Add 6 to the simplified fraction: 6 can be written as \( \frac{162}{27} \) (since \( 6 \times 27 = 162 \)). Add to get \( \frac{31}{27} + \frac{162}{27} = \frac{193}{27} \).

Step 7: Simplify the Final Answer

Simplify if possible. \( \frac{193}{27} \) is already fully simplified, so it is the final answer.

Key Concepts

Fraction Simplification

Fraction simplification is all about reducing a fraction to its simplest form. When fractions are simple, they are easier to understand and work with. A fraction consists of a numerator (the top number) and a denominator (the bottom number). To simplify a fraction, you must find the greatest common divisor (GCD) of the numerator and the denominator and divide both by this number.
For example, let's take the fraction \( \frac{3}{9} \). The GCD of 3 and 9 is 3. So, if you divide both the numerator and the denominator by 3, you will simplify \( \frac{3}{9} \) to \( \frac{1}{3} \).

• Identify the GCD of the numerator and denominator.
• Divide both numbers by their GCD to simplify.

Simplification helps in calculating and comparing fractions more easily.

Powers of Fractions

Working with powers of fractions might seem tricky at first, but it's quite straightforward once you get the hang of it. **Taking the power of a fraction** means multiplying the fraction by itself a certain number of times.
Let's consider \( \left( \frac{1}{3} \right)^2 \). Here, you multiply \( \frac{1}{3} \) by itself:

• \( \frac{1}{3} \times \frac{1}{3} = \frac{1}{9} \)

For **\( \left( \frac{1}{3} \right)^3 \)**, you'd perform another multiplication:

• \( \frac{1}{9} \times \frac{1}{3} = \frac{1}{27} \)

Each step involves multiplying both the numerators together and the denominators together. This knowledge is crucial in simplifying or manipulating more complex numerical expressions.

Common Denominator

When adding or subtracting fractions, it's essential to have a common denominator; a common baseline for comparison. The denominators (bottom numbers) of the fractions need to be the same for accurate arithmetic.
To illustrate this concept, consider fractions \( \frac{1}{3} \) and \( \frac{2}{3} \). Although they have the same denominator, when dealing with different denominators, you must find a number that both denominators can divide into evenly—this is the common denominator.
For example, if you need to add \( \frac{3}{9} \) and \( \frac{2}{6} \):

• Convert \( \frac{2}{6} \) to \( \frac{3}{9} \) by finding the least common denominator, which is 9.
• Adjust \( \frac{2}{6} \) to \( \frac{3}{9} \), since \( \frac{3}{9} = \frac{3}{9} \) and now they can be summed.

Once fractions share the same denominator, you can easily add or subtract them by combining the numerators.