Root Calculator

 What Is a Root in Mathematics?

In math, a root is a number that, when multiplied by itself several times, equals a given value. This is called the nth root. For example:

n√a = b
bⁿ = a

If b multiplied by itself n times equals a, then b is the nth root of a.

 Common Types of Roots

• Square Root (n = 2)

• Cube Root (n = 3)

These are frequently used in algebra, geometry, and everyday calculations.

How to Estimate a Square Root (√a)

 Step-by-Step Method

You can estimate a square root manually using a basic approach. No advanced math is needed—just follow these steps:

1. Make a guess (b)

2. Divide a by b to get c

3. Find the average: (b + c) / 2

4. Use this new value as the next guess and repeat step 2 until it’s accurate

 Example: Estimate √27 to 3 Decimal Places

• Start with a guess: 5.125

• 27 ÷ 5.125 = 5.268

• Average: (5.125 + 5.268) / 2 = 5.197

• 27 ÷ 5.197 = 5.195

• Average: (5.195 + 5.197) / 2 = 5.196

• 27 ÷ 5.196 = 5.196

•  Final Answer: √27 ≈ 5.196

How to Estimate an Nth Root (n√a)

 Steps to Manually Find an Nth Root

Estimating higher roots is similar but needs a slight adjustment in the formula:

1. Guess a starting number (b)

2. Divide a by bⁿ⁻¹ to get c

3. Update your guess using: [(b × (n – 1)) + c] / n

4. Repeat the process until the value is accurate

 Example: Estimate 8th Root of 15 (8√15) to 3 Decimal Places

• First guess: 1.432

• 15 ÷ (1.432⁷) = 1.405

• Average: (1.432 × 7 + 1.405) / 8 = 1.388

• 15 ÷ (1.388⁷) = 1.403

• Average: (1.403 × 7 + 1.388) / 8 = 1.402

• Final Answer: 8√15 ≈ 1.403