Write each root using exponents and evaluate. ∜256

Verified step by step guidance

1. Identify the expression: ∜256 is the fourth root of 256.
2. Rewrite the fourth root using exponents: ∜256 = 256^{1/4}.
3. Recognize that 256 is a power of 2: 256 = 2^8.
4. Substitute the expression for 256: 256^{1/4} = (2^8)^{1/4}.
5. Apply the power of a power property: (2^8)^{1/4} = 2^{8/4}.

Key Concepts

Roots and Exponents

Roots and exponents are fundamental concepts in algebra that describe the relationship between numbers. The nth root of a number is a value that, when raised to the nth power, gives the original number. For example, the square root of 9 is 3 because 3^2 = 9. This relationship can also be expressed using fractional exponents, where the nth root of a number 'a' is represented as a^(1/n).

Evaluating Roots

Evaluating roots involves calculating the value of a number when raised to a specific root. In the case of ∜256, we are looking for a number that, when raised to the fourth power, equals 256. This process often requires knowledge of perfect squares and cubes, as well as the ability to simplify expressions to find the root efficiently.

Perfect Powers

Perfect powers are numbers that can be expressed as an integer raised to an exponent. For instance, 256 is a perfect power because it can be expressed as 4^4 (4 raised to the fourth power). Recognizing perfect powers is crucial for simplifying root expressions and helps in evaluating roots more quickly, as it allows for direct computation rather than trial and error.