An Nth root calculator is a powerful mathematical tool designed to compute the nth root of a given number using y = x^(1/n) formula.
The nth root of a number x is a value that, when multiplied by itself n times, yields x. This concept extends beyond the familiar square root (2nd root) and cube root (3rd root) to encompass any positive integer n.
Consider the following examples:
- The 2nd root (square root) of 9 is 3, because 3 × 3 = 9.
- The 3rd root (cube root) of 27 is 3, because 3 × 3 × 3 = 27.
- The 4th root of 16 is 2, because 2 × 2 × 2 × 2 = 16.
Nth Root Calculator
| Number (x) | Root Degree (n) | Nth Root |
|---|---|---|
| 16 | 2 | 4 |
| 27 | 3 | 3 |
| 256 | 4 | 4 |
| 32 | 5 | 2 |
| 64 | 6 | 2 |
| 128 | 7 | 2 |
| 512 | 9 | 2 |
| 1024 | 10 | 2 |
Nth Root Formula
The nth root formula is expressed mathematically as:
y = x^(1/n)Where:
- y is the nth root of x
- x is the number for which we’re finding the root
- n is the root degree (a positive integer)
This formula is the cornerstone of nth root calculations. Let’s examine some examples to illustrate its application:
- To find the 5th root of 32: y = 32^(1/5) ≈ 2
- For the 4th root of 81: y = 81^(1/4) = 3
- Calculating the 6th root of 64: y = 64^(1/6) = 2
How do you find an nth root?
Here’s a step-by-step approach:
- Identify the number (x) and the root degree (n).
- Apply the nth root formula: x^(1/n).
- Simplify if possible.
- Use a calculator for complex calculations.
Find the 3rd root of 125.
We have x = 125 and n = 3.
Apply the formula: 125^(1/3).
This can be simplified: 5 (since 5 × 5 × 5 = 125).
What is Newton’s formula for the nth root?
The formula for finding the nth root of a number a using Newton’s method is:
x_(k+1) = (1/n) ((n - 1) x_k + a / (x_k)^(n - 1))Where:
- x_k is the current approximation
- x_(k+1) is the next approximation
- n is the root degree
- a is the number for which we’re finding the root
Newton’s method, also known as the Newton-Raphson method, is an iterative technique for finding roots of equations. It can be adapted to calculate nth roots.
Let’s use Newton’s method to approximate the cube root of 27 (3rd root of 27):
- Start with an initial guess, say x_0 = 3.
- Apply the formula: x_1 = (1/3) ((3 – 1) 3 + 27 / (3^(3 – 1))) = 3.037
- Repeat the process with x_1 as the new approximation.
- Continue until the desired accuracy is achieved.
This method converges quickly to the actual value of 3.
References
- Weisstein, Eric W. “Nth Root.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/nthRoot.html
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