Problem 15 - Algebra for College Students

Question

Evaluate each numerical expression. \(27^{\frac{4}{3}}\)

Step-by-Step Solution

Answer

The expression evaluates to 81.

Step 1: Understanding rational exponent notation

The expression given is \(27^{\frac{4}{3}}\). This expression can be understood as \((27^{\frac{1}{3}})^4\), where the denominator of the exponent (3) indicates the cube root, and the numerator (4) designates the power.

Step 2: Simplifying the cube root

First, calculate the cube root of 27, which is represented by \(27^{\frac{1}{3}}\). Since \(27 = 3^3\), \(27^{\frac{1}{3}}\) simplifies to 3.

Step 3: Raising to the power of 4

Now we raise the result from Step 2 (which is 3) to the power of 4. Calculate \(3^4\), which equals \(3 \times 3 \times 3 \times 3 = 81\).

Step 4: Conclusion

The result of evaluating the expression \(27^{\frac{4}{3}}\) is 81.

Key Concepts

Numerical Expression Evaluation

Numerical expression evaluation is the process of calculating the value of an expression that contains numbers and operations such as addition, subtraction, multiplication, division, and exponentiation. It involves carefully following the order of operations, which is often memorized using the acronym PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).
When evaluating expressions, it's crucial to simplify them step by step while maintaining the integrity of the calculations. In our example, we are tasked with evaluating the expression \(27^{\frac{4}{3}}\). The first step is to interpret the notation correctly, which involves understanding how to manage the rational exponent, as it's not a traditional whole number power.
By recognizing \(27^{\frac{4}{3}}\) as equivalent to \((27^{\frac{1}{3}})^4\), we can break down the problem into manageable parts: first calculating the cube root, followed by raising the result to the given power.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to a specific power, also referred to as an exponent. This operation is a shorthand way of indicating repeated multiplication.

• For example, \(3^4\) means multiplying the number 3 by itself four times: \(3 \times 3 \times 3 \times 3\).
• If the exponent is a fraction, like in the expression \(27^{\frac{4}{3}}\), it indicates a combination of taking roots and powers. The denominator of the fraction is the type of root you take, while the numerator is the power to which the result is raised.

In essence, in rational exponent notation, you are performing two operations: first taking a root and then raising that root to an additional power. Understanding how to parse and calculate these expressions is crucial for mastering exponentiation under rational exponents.

Cube Root

The cube root of a number is a special value that, when cubed (multiplied by itself three times), gives the original number. The cube root is often used when dealing with rational exponents involving denominators of three.
For example, to find the cube root of 27, we need to determine a value that when multiplied by itself three times results in 27.

• Since \(3 \times 3 \times 3 = 27\), the cube root of 27 is 3.
• This is represented mathematically by \(27^{\frac{1}{3}} = 3\).

Identifying the cube root is a critical part of breaking down and solving expressions with rational exponents. Once the root is calculated, it becomes easier to handle further operations such as raising to additional powers.

Simplifying Expressions

Simplifying expressions involves breaking down complex mathematical statements into their simplest form. This process can include operations such as factoring, combining like terms, and applying the rules of exponents and roots.
In the context of our original problem, we simplified the expression \(27^{\frac{4}{3}}\) by:

• Identifying the cube root of 27 as 3, using the property of exponents that allows us to treat \(27^{\frac{1}{3}}\) as the cube root of 27.
• Raising the cube root result, 3, to the power of 4, resulting in \(3^4 = 81\).

Through these steps, the expression is reduced to a simple number, making calculations straightforward and allowing us to comprehend the result more clearly. Simplification not only provides clarity but also offers insight into the structure and meaning behind mathematical expressions.