Find each root. ⁶√x^6

Verified step by step guidance

1. Recognize that the expression \( \sqrt[6]{x^6} \) represents the sixth root of \( x^6 \).
2. Recall the property of radicals and exponents: \( \sqrt[n]{a^n} = |a| \) when \( n \) is even.
3. Apply this property to the given expression: \( \sqrt[6]{x^6} = |x| \).
4. Understand that the absolute value is used because the sixth root of a number raised to the sixth power is the absolute value of the original number.
5. Conclude that the root of \( x^6 \) when taking the sixth root is \( |x| \).

Key Concepts

Radicals

Radicals are expressions that involve roots, such as square roots, cube roots, and higher-order roots. The notation ⁶√x^6 indicates the sixth root of x raised to the sixth power. Understanding how to manipulate and simplify radical expressions is essential for solving problems involving roots.

Exponents

Exponents represent repeated multiplication of a number by itself. In the expression x^6, the exponent 6 indicates that x is multiplied by itself six times. When dealing with roots, the relationship between exponents and roots is crucial, as the nth root of a number can be expressed as raising that number to the power of 1/n.

Simplifying Radicals

Simplifying radicals involves reducing a radical expression to its simplest form. For example, ⁶√x^6 simplifies to x, since the sixth root of x^6 is x itself. This concept is important for finding roots efficiently and understanding the properties of exponents and roots in algebra.