Combine Radicals Calculator

Combine Radicals Calculator

RADICAL TERM 1

+ −

RADICAL TERM 2

Result:

function calculateRadicals() { var c1 = parseFloat(document.getElementById('coeff1').value); var i1 = parseInt(document.getElementById('index1').value); var r1 = parseFloat(document.getElementById('radicand1').value); var c2 = parseFloat(document.getElementById('coeff2').value); var i2 = parseInt(document.getElementById('index2').value); var r2 = parseFloat(document.getElementById('radicand2').value); var op = document.getElementById('operation').value; var resultDiv = document.getElementById('radical-result'); var display = document.getElementById('result-display'); var explanation = document.getElementById('explanation-display'); if (isNaN(c1) || isNaN(i1) || isNaN(r1) || isNaN(c2) || isNaN(i2) || isNaN(r2)) { alert("Please enter valid numbers in all fields."); return; } resultDiv.style.display = 'block'; if (i1 === i2 && r1 === r2) { var finalCoeff; if (op === 'add') { finalCoeff = c1 + c2; } else { finalCoeff = c1 – c2; } var indexStr = (i1 === 2) ? "" : "^{" + i1 + "}"; display.innerHTML = finalCoeff + indexStr + "√(" + r1 + ")"; explanation.innerHTML = "Since both radicals have the same index (" + i1 + ") and radicand (" + r1 + "), they are 'like terms' and can be combined by " + (op === 'add' ? 'adding' : 'subtracting') + " their coefficients."; } else { var opSymbol = (op === 'add') ? "+" : "-"; var index1Str = (i1 === 2) ? "" : "^{" + i1 + "}"; var index2Str = (i2 === 2) ? "" : "^{" + i2 + "}"; display.innerHTML = c1 + index1Str + "√(" + r1 + ") " + opSymbol + " " + c2 + index2Str + "√(" + r2 + ")"; explanation.innerHTML = "These radicals cannot be combined into a single term because they do not have the same index and radicand. They must remain separate."; } }

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How to Combine Radicals

In algebra, combining radicals is very similar to combining "like terms." Just as you can add 3x + 2x to get 5x, you can add radical terms that share the same characteristics. However, there are strict rules you must follow to ensure the mathematical operation is valid.

The Two Requirements for Combining Radicals

To add or subtract two radical expressions, they must have two things in common:

• The Index: This is the degree of the root (e.g., square root has an index of 2, cube root has an index of 3).
• The Radicand: This is the number inside the radical symbol.

If either of these values differs, the terms are considered "unlike" and cannot be merged into a single radical expression.

The Formula

aⁿ√x + bⁿ√x = (a + b)ⁿ√x

Real-World Examples

Example 1: Adding Square Roots
Expression: 4√7 + 3√7
Step 1: Check if indices match (both are 2).
Step 2: Check if radicands match (both are 7).
Step 3: Add the coefficients: 4 + 3 = 7.
Result: 7√7

Example 2: Subtracting Cube Roots
Expression: 10∛5 – 2∛5
Step 1: Check if indices match (both are 3).
Step 2: Check if radicands match (both are 5).
Step 3: Subtract the coefficients: 10 – 2 = 8.
Result: 8∛5

What if the Radicals Don't Match?

If you encounter an expression like 2√3 + 5√2, you cannot combine them because the radicands (3 and 2) are different. Similarly, √5 + ∛5 cannot be combined because the indices (2 and 3) are different. In these cases, the expression is already in its simplest form.

Pro Tip: Simplify First

Sometimes radicals look different but can be combined after simplification. For example, √8 + √2 looks incompatible. However, √8 simplifies to 2√2. Therefore, 2√2 + √2 = 3√2. Always ensure your radicals are fully simplified before deciding if they can be combined.