Problem 42 - College Algebra with Modeling and Visualization

Question

Write the expression in radical notation. $$ 7^{1 / 3} $$

Step-by-Step Solution

Answer

\(7^{1/3} = \sqrt[3]{7}\)

Step 1: Understanding Exponential to Radical Conversion

To convert an expression with a fractional exponent to radical notation, recall that the exponent's numerator becomes the power, and the denominator becomes the root. In general, for an expression of the form \(a^{m/n}\), it can be rewritten in radical notation as \(\sqrt[n]{a^m}\).

Step 2: Identify Numerator and Denominator

For the given expression \(7^{1/3}\), identify the numerator which is 1, and the denominator which is 3. This implies that we have \(7\) raised to the power of 1 and the cube root.

Step 3: Apply the Radical Notation Formula

Using the formula derived in Step 1, rewrite \(7^{1/3}\). The expression in radical notation becomes \(\sqrt[3]{7^1}\) or simply \(\sqrt[3]{7}\), since raising to the power of 1 does not change the number.

Key Concepts

Fractional Exponents

Fractional exponents are a way to express powers and roots together in one compact form. They are written with a numerator and a denominator, separating these two components. In any fractional exponent, the numerator represents the power to which the base is raised, while the denominator signifies the root to be taken of that base.
For example, in the expression \(7^{1/3}\), the number 1 (the numerator) signifies that 7 is raised to the power of 1, and the number 3 (the denominator) indicates that a cube root is being taken.
Using fractional exponents is a widely used mathematical notation for simplifying complex expressions, especially when dealing with roots and powers simultaneously. They help in converting expressions between exponential and radical forms seamlessly.

Exponential to Radical Conversion

Converting from exponential to radical notation allows us to switch between these two representations of numbers easily. This conversion plays a critical role in simplifying expressions and solving equations involving roots and powers.
The rule to convert an expression with a fractional exponent \(a^{m/n}\) into radical notation is as follows:

• The denominator \(n\) becomes the root.
• The numerator \(m\) becomes the exponent inside the radical.

So, \(a^{m/n}\) can be rewritten as \(\sqrt[n]{a^m}\).
In the expression \(7^{1/3}\), applying this rule, the denominator 3 indicates the cube root, and the numerator 1 means 7 raised to the first power. Therefore, \(7^{1/3}\) is equivalent to \(\sqrt[3]{7}\), a simpler form in radical notation.

Cube Root

The cube root is a special operation in mathematics that determines which number, when multiplied by itself twice, equals the original number. It is symbolized by the radical sign with a small 3 above it: \(\sqrt[3]{}\).
Cube roots are used in various applications ranging from geometry to solving cubic equations in algebra. For example, if you have a number 8, the cube root of 8 is 2 because \(2 \times 2 \times 2 = 8\).
In radical notation, \(\sqrt[3]{7}\) signifies the cube root of 7. This indicates finding a number that, when used three times in a multiplication sequence, results in 7 itself. Cube roots provide a way to simplify expressions and can be essential in understanding more complex mathematical concepts.