General Root Calculator (nth root)
What is nth root?
The nth root of a number x, denoted as ∛x (for n=3), ∜x (for n=4), or more generally as x^(1/n), is a value that, when multiplied by itself n times, gives x. It's a generalization of the square root and cube root concepts to any positive integer n.
How to Calculate nth root
Calculating the nth root can be done using various methods, including:
- Using a scientific calculator with an nth root function
- Applying logarithms and exponents
- Using numerical methods like Newton's method for more complex cases
- For perfect nth powers, factoring and simplifying
Formula
The formula for the nth root is:
\[ y = \sqrt[n]{x} \]
Which is equivalent to:
\[ y^n = x \]
Where:
- x is the number we're finding the root of (the radicand)
- n is the degree of the root
- y is the result (the root)
Calculation Steps
- Identify the radicand (x) and the degree of the root (n)
- If using logarithms, apply the formula: y = e^(ln(x) / n)
- If using a calculator, input x and n into the nth root function
- For simple cases or perfect powers, try to identify y such that y^n = x
- Verify your result by raising it to the nth power, which should equal x
Example
Let's calculate the cube root (3rd root) of 27:
- We want to find y such that y^3 = 27
- Using a calculator or by recognizing that 3 * 3 * 3 = 27, we find that ∛27 = 3
- To verify: 3^3 = 27, which confirms our result
Therefore, ∛27 = 3
Graph of y = ∛x showing the point (27, 3)