Problem 6 - Algebra 2 and Trigonometry

Question

In \(3-37,\) express each power as a rational number in simplest form. $$ 8^{\frac{1}{3}} $$

Step-by-Step Solution

Answer

The simplest rational form of \(8^{\frac{1}{3}}\) is 2.

Step 1: Understand the Problem

We need to express the power \(8^{\frac{1}{3}}\) as a rational number. A rational number can be expressed as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(be 0\).

Step 2: Identify the Root

The power \(\frac{1}{3}\) indicates a cube root. Therefore, \(8^{\frac{1}{3}}\) is equivalent to the cube root of 8, i.e., \(\sqrt[3]{8}\).

Step 3: Calculate the Cube Root

To calculate \(\sqrt[3]{8}\), we find a number which, when multiplied by itself twice more (cubed), equals 8. Since \(2 \times 2 \times 2 = 8\), it follows that \(\sqrt[3]{8} = 2\).

Step 4: Express as a Rational Number

The result from the previous step is 2, which is already a rational number because it can be expressed as \(\frac{2}{1}\). This means the rational number form of \(8^{\frac{1}{3}}\) is simply 2.

Key Concepts

Cube Roots

Cube roots are an interesting and important concept in mathematics. When we talk about the cube root of a number, we are looking for a value that, when multiplied by itself three times, gives the original number. To express this mathematically, for a number \( x \), the cube root is written as \( \sqrt[3]{x} \) or \( x^{\frac{1}{3}} \). Cube roots can help simplify complex expressions involving cubic powers.

• Understanding the notation: \( x^{\frac{1}{3}} \) indicates we need the cube root, not simply a square root or any other root.
• Calculating cube roots: For small numbers like 8, you can determine the cube root by finding a smaller number, say 2, such that multiplying it by itself three times yields the original number, i.e., \( 2 \times 2 \times 2 = 8 \).

Remember, cube roots can render complex expressions into simpler forms, which is extremely useful in problem-solving scenarios.

Exponents

Exponents are fundamental to understanding powers in mathematics. They indicate how many times a number, known as the base, is multiplied by itself. The number \( x^{1/n} \), like \( 8^{1/3} \), showcases a particular usage of exponents for roots.
Using exponents:

• Base and exponent: In \( x^n \), \( x \) is the base and \( n \) is the exponent. Here, \( 8^{1/3} \) signifies raising 8 to the power of \( 1/3 \), indicating a cube root.
• Fractional exponents: These reflect roots. Here, \( 8^{1/3} \) is equivalent to the cube root of 8.

Understanding exponents can make working with numbers more efficient, especially for simplifying expressions with multiple layers of operations.

Simplifying Expressions

Simplifying expressions is a key skill that makes tackling math problems easier. It is about reducing expressions to their simplest, most manageable form, often using rules of algebra and the properties of exponents and roots.
Steps to simplify:

• Identify the expression: Know exactly what the math problem presents. For instance, \( 8^{1/3} \).
• Apply basic operations: Use knowledge of cube roots and exponents to transform and reduce the expression. In this case, determining that \( \sqrt[3]{8}=2 \).
• Express as a rational number: Once simplified, ensure the expression is presented as a rational number, such as rewriting 2 as \( \frac{2}{1} \).

By honing simplification skills, you can tackle complex problems more effectively and arrive at solutions faster with fewer errors.