Problem 11 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each expression. $$8^{2 / 3}$$

Step-by-Step Solution

Answer

\(8^{2/3}\) equals 4.

Step 1: Understanding the Expression

The expression given is \(8^{2/3}\). This is a fractional exponent, where the numerator of the fraction (2) is the power and the denominator (3) is the root. Thus, it means the cube root of 8 squared.

Step 2: Taking The Cube Root

First, find the cube root of 8. The cube root of a number is a value that when multiplied by itself three times equals the original number. Since \(2^3 = 8\), the cube root of 8 is 2.

Step 3: Squaring the Result

Now that we have the cube root of 8 as 2, we need to square this result. Squaring a number means multiplying it by itself. Therefore, \(2^2 = 4\).

Step 4: Final Evaluation

After squaring the result of the cube root, we have the final result as 4. Therefore, \(8^{2/3} = 4\).

Key Concepts

Cube Root

The cube root of a number is a special value that, when multiplied by itself three times, gives the original number.
Think of it as the opposite of cubing a number. The symbol for a cube root is a radical sign with a small number three.
For example, the cube root of 8 is represented as \(\sqrt[3]{8}\).

• Cube roots are important because they help us understand how quantities break down into equal parts.
• If you take the cube root of 8, you're looking for a number that when used three times in multiplication will give you the number 8.

In our original example, because \(2^3 = 8\), the cube root of 8 is 2.
This is a simple yet crucial step in solving expressions with fractional exponents.

Exponentiation

Exponentiation is a mathematical operation that involves two numbers: the base and the exponent.
The base is the number that gets multiplied, and the exponent tells you how many times to use the base in a multiplication.

• If the exponent is 2, the operation is called squaring.
• If the exponent is 3, it is called cubing, among other names for larger numbers.

For example, in the expression \(8^{2/3}\), we actually work through two operations: the cube root (as we discussed) and then raising that result to the second power (squaring).
This is an essential skill for interpreting and simplifying fractional exponents.

Radicals

A radical is represented by the symbol \(\sqrt{}\), and it indicates the root of a number.
Radicals allow us to solve equations by reversing the effect of exponentiation.

• For instance, \(\sqrt[3]{x}\) is looking for a cube root.
• Radicals can apply to any degree of root, not just square or cube roots.

When you see a radical in a math problem, it usually means you're either being asked to find a root or demonstrate your understanding of how roots counteract exponents.
In our initial problem, recognizing \(\sqrt[3]{8}\) as 2 plays a critical role in accurately evaluating the expression \(8^{2/3}\).
Understanding radicals is pivotal for mastering fractional exponents and more complex mathematics.