Write each root using exponents and evaluate. ∜-256

Verified step by step guidance

1. Recognize that the expression ∜-256 represents the fourth root of -256.
2. Express the fourth root using exponents: ∜-256 = (-256)^{1/4}.
3. Since -256 is a negative number, consider the properties of roots and exponents. The fourth root of a negative number is not a real number because even roots of negative numbers are not defined in the real number system.
4. To evaluate this in the context of complex numbers, express -256 as -1 multiplied by 256: -256 = -1 \times 256.
5. Use the property of exponents: (-1 \times 256)^{1/4} = (-1)^{1/4} \times 256^{1/4}. The term (-1)^{1/4} represents the fourth root of -1, which is a complex number.

Key Concepts

Roots and Exponents

Roots are the inverse operations of exponents. For example, the square root of a number is the same as raising that number to the power of 1/2. Similarly, the fourth root of a number can be expressed as raising that number to the power of 1/4. Understanding how to convert between roots and exponents is essential for solving problems involving radical expressions.

Negative Numbers and Even Roots

When dealing with even roots, such as the fourth root, it is important to note that the result is defined only for non-negative numbers in the real number system. For example, the fourth root of -256 does not yield a real number, as there is no real number that, when raised to an even power, results in a negative number. This concept is crucial for evaluating expressions involving even roots.

Complex Numbers

Complex numbers extend the real number system to include solutions to equations that do not have real solutions, such as the fourth root of a negative number. A complex number is expressed in the form a + bi, where 'a' is the real part and 'bi' is the imaginary part. Understanding complex numbers is essential for evaluating roots of negative numbers, as they allow for a complete solution set in mathematics.