Square Root Calculator with Radicals and Variables

This square root calculator handles expressions with radicals and variables. Enter your expression in the format √(a + b√c) or similar, and the calculator will simplify the square root for you.

How to Use This Calculator

To calculate a square root with radicals and variables:

Enter the expression inside the square root in the input field. For example, type "2 + 3√5" for √(2 + 3√5).
Click the "Calculate" button to see the simplified form.
The calculator will display the simplified radical form and a decimal approximation.
Use the "Reset" button to clear the form and start over.

The calculator handles expressions of the form √(a + b√c) where a, b, and c are numbers or variables. It will attempt to simplify the expression into a form like √d + √e.

Square Root Formula

Formula Used

The calculator uses the following approach to simplify expressions of the form √(a + b√c):

Assume the simplified form is √x + √y
Square both sides: (√x + √y)² = x + y + 2√(xy)
Set this equal to the original expression: x + y + 2√(xy) = a + b√c
Solve for x and y by matching the rational and irrational parts

The calculator implements this algorithm to find values of x and y that satisfy the equation. When exact simplification isn't possible, it provides a decimal approximation.

Worked Examples

Example 1: Simple Radical

Calculate √(2 + 2√2)

Using the formula, we find x = 1 and y = 2, so √(2 + 2√2) = √1 + √2 = 1 + √2 ≈ 2.414

Example 2: With Variables

Calculate √(a + b√c) where a=3, b=2, c=5

The calculator finds x = 1 and y = 2, so √(3 + 2√5) = √1 + √2 = 1 + √2 ≈ 3.236

Note

The calculator may not always find an exact simplified form. In such cases, it provides a decimal approximation.

Frequently Asked Questions

What types of expressions can this calculator handle?: This calculator handles expressions of the form √(a + b√c) where a, b, and c can be numbers or variables.
Does the calculator always find an exact simplified form?: No, the calculator may not always find an exact simplified form. In such cases, it provides a decimal approximation.
Can I use variables in the expressions?: Yes, you can use variables in the expressions. The calculator will attempt to simplify the expression with variables.
How accurate are the decimal approximations?: The decimal approximations are accurate to 10 decimal places.
Is there a way to visualize the results?: Yes, the calculator includes a chart that shows the relationship between the input values and the square root result.