Problem 104 - College Algebra

Question

Simplify by reducing the index of the radical. $$\sqrt[4]{x^{12}}$$

Step-by-Step Solution

Answer

\( \sqrt[4]{x^{12}} \) simplifies to \( x^3 \).

Step 1: Understand the basic property of radical

The basic property of a radical is \( \sqrt[n]{x^m} = x^{m/n} \). It means that an nth root of any number \( x^m \) can be written as a power of \( x \) where the exponent is the quotient of \( m \) divided by \( n \). Thus the given radical \( \sqrt[4]{x^{12}} \) can be written as \( x^{12/4} \).

Step 2: Simplify the exponent

Now, divide the exponent of \( x \) by the index of the radical. In our case, divide 12 by 4 to get 3. So, \( x^{12/4} \) simplifies to \( x^3 \).

Step 3: Final answer

The simplified form of the given radical \( \sqrt[4]{x^{12}} \) is \( x^3 \).