Problem 22 - Brief Applied Calculus

Question

Evaluate each expression without using a calculator. $$ 27^{2 / 3} $$

Step-by-Step Solution

Answer

The expression evaluates to 9.

Step 1: Understand the Expression

The expression given is \(27^{2/3}\). Here, the fraction \(2/3\) in the exponent denotes a two-step process: first taking the cube root (denominator), and then squaring the result (numerator).

Step 2: Simplify the Cube Root

Start by finding the cube root of 27. The cube root asks, "Which number multiplied by itself three times gives 27?" The answer is 3, since \(3 \times 3 \times 3 = 27\). Thus, \(27^{1/3} = 3\).

Step 3: Apply the Numerator of the Exponent

Now, take the result from Step 2 and raise it to the power of 2. This means squaring 3: \(3^2 = 9\).

Step 4: Combine Results

Combine the results of the previous steps. We determined that \(27^{2/3} = (27^{1/3})^2 = 3^2 = 9\). Therefore, the expression evaluates to 9.

Key Concepts

Cube root

The cube root is an important mathematical concept, often used in simplifying expressions with exponents. When we talk about the cube root of a number, we're asking for the value that, when multiplied by itself three times, results in the original number. For instance, finding the cube root of 27 means finding a number that satisfies the equation:

• \( x^3 = 27 \)

Here, the answer is 3 because:

• \( 3 \times 3 \times 3 = 27 \)

Knowing the cube root is vital because it simplifies expressions involving roots and powers. In our original exercise with the expression \((27)^{2/3}\), identifying the cube root of 27 as 3 was the first step to solving the problem without a calculator. This simplification reduces the complexity of the expression and allows further calculation, such as squaring the result, which is especially convenient in solving exponentiation problems step by step.

Fractional exponents

Fractional exponents may seem daunting, but they hold a straightforward meaning once understood. In mathematics, an exponent can represent a fraction as well as an integer.

• The denominator of the fraction dictates the root to be taken.
• The numerator indicates the power to which the root is raised.

For example, in the expression \( 27^{2/3} \):

• The denominator \(3\) signals the need to find the cube root of 27.
• After finding the cube root, the numerator \(2\) indicates that the result should be squared.

Let's break them down further.
The expression first transforms to \( 27^{1/3} = 3 \), as explained in the cube root section. Then we raise the result to the power of 2, symbolized by \( 3^2 \), to ultimately get 9. Understanding these fractional exponents unlocks a systematic approach to simplifying and calculating expressions by efficiently breaking them down into manageable operations.

Simplifying expressions

Simplifying expressions is a crucial skill in mathematics, enabling the reduction of complex expressions to their simplest form. This skill not only makes computation easier but also enhances understanding.

• Start by identifying the type of expression: polynomial, rational, exponential, etc.
• For expressions with exponents, follow the order dictated by fractional exponents or integer powers.

In our example of \( 27^{2/3} \), simplifying it involves decomposing the work into two steps based on the fractional exponent:

• First, simplify by taking the cube root to get a base number, here, \(3\).
• Then apply the remaining power, which in this case is squaring the base, giving \( 9 \).

These steps showcase that simplifying expressions involves understanding and applying mathematical operations efficiently.
This often includes breaking down a problem into simpler operations that reveal the underlying structure. Simplification helps provide clarity and structure in solving mathematical problems, easing the load on memory and reducing the risk of error.