Problem 15 - A Graphical Approach to College Algebra

Question

Simplify each expression. Assume that all variables represent positive real numbers. $$8^{2 / 3}$$

Step-by-Step Solution

Answer

The simplified form of \(8^{2/3}\) is 4.

Step 1: Understand the Expression

The expression given is \(8^{2/3}\), which is a number to a fractional exponent. This type of exponent can be simplified using the property of exponents: \(a^{m/n} = \sqrt[n]{a^m}\).

Step 2: Rewrite Using Radical Form

Rewrite the expression \(8^{2/3}\) in radical form. Based on the exponent property, \(8^{2/3}\) can be written as \(\sqrt[3]{8^2}\).

Step 3: Simplify Inside the Radical

First, calculate \(8^2\), which equals 64. So our expression is now \(\sqrt[3]{64}\).

Step 4: Calculate the Cube Root

Find the cube root of \(64\), which is \(\sqrt[3]{64}\). Since \(4^3 = 64\), the cube root of 64 is 4.

Key Concepts

Exponent Properties

Fractional exponents, like the one in the expression \(8^{2/3}\), can initially look confusing, but they actually hold a special property. These are part of what we call **Exponent Properties**. Here's a quick guide to help you understand what’s happening:

• In fractional exponents, the numerator determines the power, while the denominator determines the root.
• In the form \(a^{m/n}\), \(m\) is the power to which the base \(a\) is raised, and \(n\) is the type of root we take.
• This property allows us to convert between exponent and radical forms, such that \(a^{m/n} = \sqrt[n]{a^m}\).

In our exercise, \(8^{2/3}\) means to square the 8 and then take the cube root, helping us simplify otherwise complex expressions neatly.

Radical Expressions

When dealing with radical expressions, such as \(\sqrt[3]{64}\), conversion from an exponent to a radical can simplify the process significantly. Here's what you should keep in mind:

• A radical expression involves roots. In this case, \(\sqrt[3]{}\) represents a cube root.
• The term inside the radical, here \(64\), can often be made less intimidating by knowing exponent properties.
• The process involves simplifying what's inside the radical before finding the root, making calculations more straightforward.

So, in the example \(\sqrt[3]{8^2}\), we first computed \(8^2 = 64\). By breaking it down step-by-step, dealing with radicals becomes much more accessible.

Cube Roots

Cube roots, represented as \(\sqrt[3]{x}\), are a specific and vital part of radical expressions. Understanding them can make a big difference:

• The cube root of a number \(x\) is a value that, when multiplied by itself three times, gives \(x\).
• For example, \(\sqrt[3]{64} = 4\) because \(4^3 = 64\).
• Recognizing perfect cubes (e.g., 8, 27, 64, 125) can speed up finding cube roots without needing a calculator.

Cube roots are just another way to simplify expressions via exponentiation, and mastering them can ease manipulations of expressions like our original \(8^{2/3}\).