Radical Equation Calculator

Radical Equation Calculator:

Use our Radical Equation Calculator to solve equations containing radical expressions step by step. Simply input your equation, get detailed solutions, and learn how to eliminate radicals from your equations.

What Is A Radical Equation?

“A radical equation is an equation that contains variables inside a radical, such as a square root, cube root, or higher-order root.”

For example, √(x + 1) = 5 is a radical equation.

To understand radical equations, it's necessary to know the basic components of a radical equation.

The term radical comes from the Latin word 'radix', which means 'root'. In mathematics, radicals are used to represent roots of numbers. It is defined as:

Standard Form of Radical Equation:

The simplest form of the radical expression is given as:

n√a

Where:

√ = Radical symbol
a = Radicand (the number or expression inside the radical)
n = Index (the degree of the root)

How To Solve A Radical Equation?

Rearrange the equation to isolate the radicals on one side
To eliminate the radicals, apply the same power to both sides of the equations
Rearrange the equation and solve the resulting expressions
Verify the answer by substituting it into the original equation

Example:

Solve the equation: √(x + 5) = x - 1

Solution:

Step 1: Isolate the Radical

In this case, the radical is already isolated on the left side of the equation.

Step 2: Square Both Sides

To eliminate the radical, square both sides of the equation:

(√(x + 5))² = (x - 1)²

This simplifies to:

x + 5 = x² - 2x + 1

Step 3: Rearrange the Equation

Bring all terms to one side to form a quadratic equation:

x² - 2x + 1 - x - 5 = 0

x² - 3x - 4 = 0

Step 4: Solve the Quadratic Equation

By using factoring or completing the square, we can solve this formula:

(x - 4)(x + 1) = 0

This gives us two potential solutions:

x = 4

x = -1

How To Use The Radical Equation Calculator?

✔️ Input Your Equation:

Type or copy/paste the radical equation into the text box provided
If you have a screenshot or image of the equation, you can input it using the camera or select an image from your device

✔️ Solve the Equation:

Click the "CALCULATE" button to process the equation

✔️ View Results:

This radical equation calculator will present the step-by-step solution

✔️ Additional Features:

A reset button allows you to erase the current input and start with a new equation
Load Example button shows how to input equations or provide sample problems to work with

What Are The Properties of Radicals?

1. Product Property of Radicals:

It is stated that the square root of the product of two numbers equals the product of their square roots.

ⁿ a x ⁿ b = ⁿ a•b

Example: √4 x √9 = √36 = 6

2. Quotient Property of Radicals:

The quotient of the square root of two numbers is the same as the square root of their quotient

ⁿ a / ⁿ b = ⁿ a/b , b ≠ 0

Example: √25 / √4 = √25/4 = 5/2

3. Power Property of Radicals:

A radical can be expressed as a fractional exponent, with the power of the base in the numerator and the index of the radical in the denominator.

ⁿ a ^m = a^(m/n)

Example:

³ (8)² = 8^⅔ = 4

4. Radical of A Radical:

An expression refers to a radical of radical in which one radical is contained within another.

^m n a = (m*n) ³ a

Example:

√√16 = 4√16 = 2

5. Addition and Subtraction of Like Radicals:

It is a method of merging radicals that have the same index and the same value under the radical.

^a b + ^c b = (a + c) ³ b

Example:

³ 2 + ⁵ 2 = (3 + 5) ³ 2

6. Multiplication of Radicals:

Radical multiplication is the process of combining two or more radical expressions through multiplication.

√a x √a = a

Example:

√5 x √5 = 5

7. Simplifying a Radical:

Radical simplification is the process that involves transforming a radical expression into a simple or alternative form.

√(a² * b) = ^a b

Example:

√36x² = 6x

These are the properties that help to simplify and solve radical equations. To verify your work or solve complicated calculated problems, try our radical equation calculator. It handles a wide range of radical equations and provides you with step-by-step solutions.

FAQ’s:

What Is The Difference Between A Radical Expression and A Radical Equation?

The radical expression does not have an equal sign. In comparison, a radical equation contains an equal sign separating two expressions.

What Makes A Radical Have No Solution?

A radical equation has no solution when: The radicand (the number inside the radical) is negative for an even root (e.g.,√(−1) is not a real number) The solution reaches an extraneous root, which means when you put it back into the original equation, it does not satisfy the equation

What Happens If The Index of The Radical Is Even or Odd?

The index of the radical helps to determines whether the root is defined for all real numbers or restricted:

If the value of the index is even (n =2, 4, 6…):

The variable a inside the root must be non-negative (a ≥ 0) For example: √9 = valid, √-9 = not valid

If the value of the index is odd (n = 1, 3, 5, 7....):

The radicand can be any real number(positive or negative)

³ 8 =2

References:

LibreTexts Mathematics: Radical Equations, How to solve radical equations.

Khan Academy: Radical equations & functions.

Lumen Learning: Solve Radical Equations, Identify a Radical Equation with No Solutions or Extraneous Solutions.