Problem 3 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the expression. $$ \left(\frac{2}{3}\right)^{3} $$

Step-by-Step Solution

Answer

Evaluating the expression \(\left(\frac{2}{3}\right)^{3}\), we raise both the numerator and the denominator to the power of 3: \(\frac{2^{3}}{3^{3}}\). We calculate \(2^{3}=8\) and \(3^{3}=27\), and write our final expression as \(\frac{8}{27}\). This cannot be further simplified, so the evaluated expression is \(\boxed{\frac{8}{27}}\).

Step 1: Identify the base and the exponent

In the given expression \(\left(\frac{2}{3}\right)^{3}\), we have a fraction \(\frac{2}{3}\) as the base and 3 as the exponent.

Step 2: Raise the fraction to the power of 3

To raise the fraction \(\frac{2}{3}\) to the power of 3, we need to raise both the numerator and the denominator to the power of 3. Thus, we have: \[ \left(\frac{2}{3}\right)^{3} = \frac{2^{3}}{3^{3}} \]

Step 3: Evaluate the exponentials

Calculate the value of each exponential. We have: \(2^{3} = 2 \times 2 \times 2 = 8\) \(3^{3} = 3 \times 3 \times 3 = 27\)

Step 4: Write the final expression

Replace the exponential expressions in the fraction with their calculated values: \[ \frac{2^{3}}{3^{3}} = \frac{8}{27} \]

Step 5: Simplify, if possible

In this case, \(\frac{8}{27}\) cannot be further simplified, as there are no common factors between the numerator and the denominator. So, the expression is already in its simplest form. Therefore, the evaluated expression is \(\boxed{\frac{8}{27}}\).

Key Concepts

Fraction Exponents

Understanding fraction exponents is a key aspect of mastering exponentiation, a fundamental concept in algebra. Fraction exponents represent powers that are fractions, rather than whole numbers. In an expression such as \( a^{\frac{m}{n}} \), we interpret it as the nth root of \( a \) raised to the mth power. This might look complicated, but it's tied to a more comprehensible idea: roots. For instance, \( a^{\frac{1}{2}} \) is simply the square root of \( a \), and \( a^{\frac{2}{3}} \) means you take the cube root of \( a \) and then square the result.

In the case of the provided exercise, we had a whole number as the exponent \( \left(\frac{2}{3}\right)^3 \). While it's not a fraction exponent, understanding how to manipulate fraction powers can help with exercises like these where the base is a fraction. When raising a fraction to a power, the rule is to apply the exponent to both the numerator and the denominator independently, such as \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \), which is what we did in Step 2 of the solution.

Simplifying Expressions

Simplifying expressions is all about making mathematical problems easier to read and understand, by removing complexity without changing the value. This process can involve combining like terms, reducing fractions, or applying rules of exponents. The goal is to make the expression as compact and elegant as possible, while ensuring it still represents the same quantity.

In the exercise, after applying the exponent to the fraction, we obtained \( \frac{2^3}{3^3} \), and then we simplified the expression by actually calculating the exponentiations, which resulted in \( \frac{8}{27} \). The last step of our simplification was to check if the fraction could be reduced further. Since 8 and 27 do not share any common factors other than 1, the expression cannot be simplified any further. Therefore, \( \frac{8}{27} \) is already in its simplest form. This is an essential skill, as working with simpler expressions can save time in calculations and reduce the likelihood of making errors.

Mathematical Notation

Mathematical notation is the system of symbols and signs used to write equations and expressions. It's like the language of mathematics; being able to 'speak' this language is crucial for understanding and solving mathematical problems. Each symbol has a specific meaning, and the rules for combining symbols govern how we interpret the expression. For example, in our exercise, the parentheses \( ( ) \) indicate the fraction should be considered as a whole before applying the power, and the caret symbol '^' indicates exponentiation.

Appropriate use of mathematical symbols helps us to clearly convey what we are trying to compute. It's also important to write the final answer in a commonly accepted form, using proper notation, so it can be easily understood by others. The box around the final answer \( \boxed{\frac{8}{27}} \) demonstrates that this is the final simplified result of the given expression. Developing a strong grasp of mathematical notation will not only help in the understanding of current problems but is also fundamental for grasping more complex concepts in the future.