Problem 18 - Algebra 2

Question

Write each expression in radical form. $$ 4^{\frac{1}{3}} $$

Step-by-Step Solution

Answer

\(\sqrt[3]{4}\)

Step 1: Identify the Problem

The problem asks us to write an expression, specifically \(4^{\frac{1}{3}}\), in radical form.

Step 2: Understand Rational Exponents

To convert expressions with rational exponents to radical form, remember that an expression \(a^{\frac{m}{n}}\) can be rewritten as the n-th root of \(a^m\), or \(\sqrt[n]{a^m}\).

Step 3: Apply the Rule of Radical Form

In our problem, the expression is \(4^{\frac{1}{3}}\). Here, \(a = 4\), \(m = 1\), and \(n = 3\), so it can be rewritten in radical form as \(\sqrt[3]{4^1}\).

Step 4: Simplify the Expression

Since \(4^1 = 4\), the expression simplifies further to \(\sqrt[3]{4}\).

Key Concepts

Rational Exponents

When working with exponents in mathematics, you often encounter rational exponents. These are exponents that are fractions, such as \(\frac{1}{3}\) instead of whole numbers. Rational exponents are a compact way to represent roots and powers in one notation.

• The denominator of the fraction tells you which root to take. For example, in \(4^{\frac{1}{3}}\), the number 3 means you're looking for the cube root.
• The numerator indicates the power you raise it to before taking the root, though it is often 1.

To convert a rational exponent into a more familiar format, you can use the expression \(a^{\frac{m}{n}}\) as \(\sqrt[n]{a^m}\). Understanding this will make transitioning between fractional exponents and radical forms much simpler.

Radical Form

Radical form is a way to express roots in mathematics. Instead of using exponents, you use the radical sign (\(\sqrt{}\)) to denote which root of the number you're taking.

• The attribute inside the radical sign is called the radicand. For \(\sqrt[3]{4}\), 4 is the radicand.
• The small number outside, above the radical sign is the index. It indicates the degree of the root. In \(\sqrt[3]{4}\), 3 is the index, showing it's a cube root.

This notation is especially handy when simplifying or manipulating expressions in equations and problems. Converting a rational exponent, like \(4^{\frac{1}{3}}\), to its radical form (\(\sqrt[3]{4}\)) makes the mechanical process of solving the problem more straightforward and less prone to error.

Cube Roots

Cube roots are specific types of roots where the index, the small number above the radical sign, is 3. This means you're finding a number that, when multiplied by itself three times, equals the radicand.

• The cube root of 8 is 2 because \(2 \times 2 \times 2 = 8\).
• For the expression \(\sqrt[3]{4}\), we are identifying a value, which when cubed, results in 4.

Understanding cube roots is crucial for mastering higher-level math concepts, including anything involving dimensional analysis or growth over three dimensions. They're often used in geometry and physics to solve problems related to volumes and rates of change."