Problem 96 - College Algebra

Question

In Exercises \(95-102,\) simplify by reducing the index of the radical. $$\sqrt[4]{7^{2}}$$

Step-by-Step Solution

Answer

The simplified expression for \(\sqrt[4]{7^{2}}\) is \(\sqrt{7}\).

Step 1: Understand the Elements of the Radical

In the given radical \(\sqrt[4]{7^{2}}\), the number 4 is the index (root), 7 is the radicand (base number) and 2 is the exponent of the base number. The task is to simplify the radical by reducing the index.

Step 2: Apply Properties of Radicals

The properties of radicals state that \(\sqrt[n]{a^m} = a^{m/n}\). This property will be used to simplify the given radical.

Step 3: Simplify the Expression

As per the property from step 2, we can rewrite the expression as \(\sqrt[4]{7^{2}} = 7^{2/4}\). Simplify the exponent by reducing the fraction to its lowest term, so \(7^{2/4} = 7^{1/2}\). The exponent 1/2 corresponds to the square root in radical notation.

Step 4: Write the Final Answer

So, the simplified form of \(\sqrt[4]{7^{2}}\) is \(7^{1/2}\), or in radical notation, \(\sqrt{7}\).

Key Concepts

Radical Notation

Radical notation is a way of representing roots of numbers and expressions, such as square roots, cube roots, and so on. It consists of a radical symbol \( \sqrt{} \) and has two parts: the radicand and the index. The radicand is the number or expression inside the radical sign, while the index, which is not always shown, indicates the degree of the root. When the index is 2, it represents a square root, and it is typically omitted because square roots are so common.

For example, in the radical expression \( \sqrt[4]{7^{2}} \) given in our exercise, \(7^{2}\) is the radicand, while the number 4 is the index, indicating a fourth root. By understanding the notation, one can then apply various properties of radicals to simplify them.

Properties of Radicals

The properties of radicals are rules that help us handle expressions under radical signs. One key property allows us to convert between radical notation and exponential notation: \( \sqrt[n]{a^m} = a^{m/n} \). In this manner, exponents can be manipulated more easily, making simplification possible.

Another property states that multiplying two radicals with the same index is equivalent to taking the root of their product: \( \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab} \). There's also a similar property for division. It's essential to remember that these properties apply when the radicands are non-negative to ensure we are working within the realm of real numbers. Students can use these properties to simplify radical expressions and solve equations involving radicals.

Exponent Reduction

Exponent reduction is a process where you simplify an expression with exponents, especially within radicals, by dividing the exponent by the index of the root. This process ties in with the properties of radicals and is a fundamental aspect of simplifying complex radical expressions.

In the exercise \( \sqrt[4]{7^{2}} \) we're faced with, we reduce the exponent 2 by the index 4, ending up with \( 7^{2/4} \). Simplifying the fraction \( 2/4 \) to \( 1/2 \) gets us \( 7^{1/2} \), which is more straightforward to interpret. This new exponent \( 1/2 \) corresponds to the square root, which leads to the final expression \( \sqrt{7} \). This method clears up seemingly complex radicals, revealing simpler forms and often making further algebraic manipulation far more manageable.