%5E1%2F4.jpeg "(16/81)^1/4")
Understanding (16/81)^(1/4)
The expression (16/81)^(1/4) represents the fourth root of the fraction 16/81. Let's break down how to solve this:
Understanding Exponents and Roots
- Exponent: An exponent indicates how many times a base number is multiplied by itself. For example, 2^3 means 2 * 2 * 2 = 8.
- Root: A root is the opposite of an exponent. The nth root of a number is the value that, when multiplied by itself 'n' times, equals the original number. For example, the square root of 9 (√9) is 3 because 3 * 3 = 9.
Solving (16/81)^(1/4)
- Simplify the fraction: 16/81 can be expressed as (2^4)/(3^4).
- Apply the exponent rule: (a/b)^n = a^n/b^n. Therefore, (2^4/3^4)^(1/4) = (2^4)^(1/4) / (3^4)^(1/4)
- Simplify the exponents: (2^4)^(1/4) = 2^(4 * 1/4) = 2^1 = 2 Similarly, (3^4)^(1/4) = 3^(4 * 1/4) = 3^1 = 3
Therefore, (16/81)^(1/4) = 2/3.
Conclusion
The fourth root of 16/81 is 2/3. This problem demonstrates the use of exponent and root rules to simplify mathematical expressions.