Problem 11 - A Graphical Approach to College Algebra

Question

Evaluate each expression. Do not use a calculator. $$8^{2 / 3}$$

Step-by-Step Solution

Answer

8^{2/3} = 4

Step 1: Understand the Expression

The expression to evaluate is \(8^{2/3}\). This means we need to find the number that when raised to the power of \(3\) gives \(8\), and then take that result and raise it to the power of \(2\). Essentially, we are dealing with a fractional exponent.

Step 2: Evaluate the Denominator of the Exponent

The exponent \(2/3\) can be broken down into two parts. First, we deal with the denominator \(3\), which means to take the cube root of \(8\). Since \(2^3 = 8\), the cube root of \(8\) is \(2\).

Step 3: Evaluate the Numerator of the Exponent

Now, take the result from Step 2, which is \(2\), and raise it to the power of the numerator, which is \(2\) in this case. So, calculate \(2^2 = 4\).

Step 4: Combine the Results

The result of \(8^{2/3}\) is the answer from Step 3, which is \(4\). Therefore, \(8^{2/3} = 4\).

Key Concepts

Cube Root

The cube root is a special mathematical operation that finds the number which, when multiplied by itself three times, gives the original number. Think of it as the opposite of cubing a number.
For example, to find the cube root of 8, you need to determine what number multiplied by itself three times equals 8. In this case, the answer is 2, because \(2 \times 2 \times 2 = 8\).
Understanding cube roots is essential when evaluating expressions with fractional exponents, as they often require taking roots as part of the operation. If you see an expression like \(x^{1/3}\), it signifies the cube root of \(x\).

• The cube root of 27 is 3, because \(3^3 = 27\).
• The cube root of 64 is 4, because \(4^3 = 64\).

Knowing how to determine cube roots mentally can greatly aid in solving problems without a calculator.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The operation \(b^n\) means multiplying the base \(b\) by itself \(n\) times.
This operation helps in expressing repeated multiplication concisely. For example, \(2^3 = 2 \times 2 \times 2 = 8\). With fractional exponents such as \(8^{2/3}\), you are dealing with roots and powers wrapped into one expression.
The exponentiation process for fractional exponents involves two steps:

• Taking the root as indicated by the denominator.
• Raising the result to the power indicated by the numerator.

This dual-step process helps simplify expressions to a more manageable form, as seen in the steps for solving \(8^{2/3}\), where the final result is 4.

Mathematical Expression Evaluation

Evaluating mathematical expressions is the process of performing operations in the correct sequence to find a single value. This involves following the established order of operations, often remembered through the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
In the context of the expression \(8^{2/3}\), our focus is on exponents, specifically fractional exponents.
To evaluate \(8^{2/3}\), start by understanding the fractional exponent:

• The denominator (3) tells us to take the cube root of 8, resulting in 2.
• The numerator (2) then indicates that we raise 2 to the power of 2, resulting in 4.

This is a systematic process where you break down complex expressions into simpler steps, making them easier to solve. Mastering this skill allows you to solve a variety of mathematical problems more efficiently.