Simplify and combine radicals with different coefficients and indices.
Term 1
√
+
−
Term 2
√
Result:
How to Add and Subtract Radicals
To add or subtract radicals, they must be "like radicals." This means they must have the same index (the small number outside the root) and the same radicand (the number inside the root).
Step 1: Simplify Each Radical
Before adding, you must simplify each radical by pulling out perfect powers. For example, in a square root, you look for perfect squares like 4, 9, 16, or 25.
Example: √8 = √(4 × 2) = 2√2
Step 2: Identify Like Radicals
Once simplified, check if the radicands and indices match. If they do, you can add or subtract the coefficients (the numbers in front).
Example: 2√2 + 3√2 = (2 + 3)√2 = 5√2
The Rule Formula
aⁿ√x + bⁿ√x = (a + b)ⁿ√x
If the radicals are not "like" and cannot be simplified to become like, the expression cannot be combined into a single term and remains as is.
Example Calculation
Suppose you want to calculate √12 + √27:
Simplify √12: √(4 × 3) = 2√3
Simplify √27: √(9 × 3) = 3√3
Combine: 2√3 + 3√3 = (2 + 3)√3 = 5√3
function simplifyRadicalLogic(c, i, r) {
var coeff = parseFloat(c);
var index = parseInt(i);
var radicand = parseInt(r);
if (isNaN(coeff) || isNaN(index) || isNaN(radicand)) return null;
if (radicand = 2; j–) {
var power = Math.pow(j, index);
while (currentRadicand % power === 0) {
currentCoeff *= j;
currentRadicand /= power;
}
}
return {
c: currentCoeff,
i: index,
r: currentRadicand
};
}
function formatRadical(res) {
if (res === "Imaginary") return "Complex Number";
if (res.r === 1) return res.c.toString();
if (res.r === 0) return "0";
var coeffStr = res.c === 1 ? "" : (res.c === -1 ? "-" : res.c);
var indexStr = res.i === 2 ? "" : "" + res.i + "";
return coeffStr + indexStr + "√" + res.r;
}
function calculateRadicals() {
var c1 = document.getElementById("coeff1").value;
var i1 = document.getElementById("index1").value;
var r1 = document.getElementById("radicand1").value;
var c2 = document.getElementById("coeff2").value;
var i2 = document.getElementById("index2").value;
var r2 = document.getElementById("radicand2").value;
var op = document.getElementById("operation").value;
var term1 = simplifyRadicalLogic(c1, i1, r1);
var term2 = simplifyRadicalLogic(c2, i2, r2);
var resultArea = document.getElementById("rad-result-area");
var output = document.getElementById("rad-final-output");
var steps = document.getElementById("rad-step-details");
if (!term1 || !term2) {
alert("Please enter valid numbers.");
return;
}
resultArea.style.display = "block";
var t1Formatted = formatRadical(term1);
var t2Formatted = formatRadical(term2);
var opChar = op === "add" ? "+" : "-";
var finalResult = "";
var stepText = "Simplified Terms: " + t1Formatted + " " + opChar + " " + t2Formatted;
// If indices and radicands match, we can combine
if (term1.i === term2.i && term1.r === term2.r) {
var combinedCoeff;
if (op === "add") {
combinedCoeff = term1.c + term2.c;
} else {
combinedCoeff = term1.c – term2.c;
}
var finalObj = { c: combinedCoeff, i: term1.i, r: term1.r };
finalResult = formatRadical(finalObj);
stepText += "Since indices and radicands match, we combine the coefficients.";
} else {
finalResult = t1Formatted + " " + opChar + " " + t2Formatted;
stepText += "Indices or radicands do not match. These terms cannot be combined further.";
}
output.innerHTML = finalResult;
steps.innerHTML = stepText;
}