Rationalized Expression:Enter values to calculate.
Understanding Expression Rationalization
Expression rationalization is a fundamental technique in algebra used to eliminate radicals (like square roots, cube roots, etc.) from the denominator of a fraction. This process makes the expression easier to work with, simplifies calculations, and is often a required step in solving algebraic problems and in calculus.
Why Rationalize?
Simplification: Fractions with radical denominators are often considered "unsimplified." Rationalizing results in a more standard and manageable form.
Calculation: It simplifies computations, especially when dealing with approximations or further algebraic manipulations.
Further Analysis: In calculus, rationalizing denominators can be crucial for finding limits, derivatives, and integrals.
How It Works: The Math Behind It
The core principle relies on multiplying the numerator and denominator by a carefully chosen expression, essentially multiplying by 1 (in the form of <chosen_expression>/<chosen_expression>), so the overall value of the fraction doesn't change. The 'chosen expression' is designed to eliminate the radical in the denominator.
Case 1: Denominator is a Single Radical (e.g., a/sqrt(b))
To rationalize a denominator like sqrt(b), you multiply both the numerator and the denominator by sqrt(b):
(a / sqrt(b)) * (sqrt(b) / sqrt(b)) = (a * sqrt(b)) / (sqrt(b) * sqrt(b)) = (a * sqrt(b)) / b
The radical is now gone from the denominator.
Case 2: Denominator is a Binomial with a Radical (e.g., a / (c + sqrt(d)))
For denominators involving binomials with square roots, we use the "difference of squares" pattern: (x - y)(x + y) = x^2 - y^2.
If the denominator is c + sqrt(d), we multiply the numerator and denominator by its conjugate, which is c - sqrt(d):
The denominator becomes: c^2 - (sqrt(d))^2 = c^2 - d. This eliminates the radical.
Similarly, if the denominator is c - sqrt(d), we multiply by the conjugate c + sqrt(d).
Using This Calculator
Enter the expression for the numerator and the denominator in the fields provided. The calculator will attempt to rationalize the denominator based on these common forms. For complex expressions, manual algebraic manipulation might still be necessary.
Note: This calculator is designed for basic rationalization involving single square roots or binomials with square roots. For higher-order roots or more complex algebraic structures, manual methods are recommended.
function rationalizeExpression() {
var numeratorInput = document.getElementById("numerator").value.trim();
var denominatorInput = document.getElementById("denominator").value.trim();
var outputElement = document.getElementById("output");
outputElement.textContent = "Calculating…";
if (numeratorInput === "" || denominatorInput === "") {
outputElement.textContent = "Please enter both numerator and denominator.";
return;
}
// Basic parsing for common forms like 'sqrt(x)' and simple numbers
// This is a simplified parser. For a robust solution, a dedicated math parsing library would be needed.
var num = numeratorInput;
var den = denominatorInput;
var rationalizedNum = "";
var rationalizedDen = "";
// — Logic for Single Radical Denominator —
// Looks for patterns like "sqrt(x)" or "a*sqrt(x)" or "sqrt(x)*a" in denominator
var sqrtMatch = den.match(/sqrt\((\d+(\.\d+)?)\)/);
var coefficientMatch = den.match(/(\d+(\.\d+)?)\s*\*\s*sqrt\((\d+(\.\d+)?)\)/);
var sqrtCoefficientMatch = den.match(/sqrt\((\d+(\.\d+)?)\)\s*\*\s*(\d+(\.\d+)?)/);
if (sqrtMatch && sqrtMatch[0] === den) { // Exactly "sqrt(b)"
var radicalPart = "sqrt(" + sqrtMatch[1] + ")";
var multiplier = radicalPart;
rationalizedNum = "(" + num + ") * " + multiplier;
rationalizedDen = "(" + radicalPart + ") * " + radicalPart + " = " + sqrtMatch[1];
} else if (coefficientMatch && coefficientMatch[0] === den) { // Exactly "a * sqrt(b)"
var coeff = parseFloat(coefficientMatch[1]);
var radicalPart = "sqrt(" + coefficientMatch[3] + ")";
var multiplier = coeff + " * " + radicalPart;
rationalizedNum = "(" + num + ") * " + multiplier;
rationalizedDen = "(" + multiplier + ") * " + radicalPart + " = " + (coeff * coeff) + " * " + coefficientMatch[3];
} else if (sqrtCoefficientMatch && sqrtCoefficientMatch[0] === den) { // Exactly "sqrt(b) * a"
var coeff = parseFloat(sqrtCoefficientMatch[3]);
var radicalPart = "sqrt(" + sqrtCoefficientMatch[1] + ")";
var multiplier = radicalPart + " * " + coeff;
rationalizedNum = "(" + num + ") * " + multiplier;
rationalizedDen = "(" + multiplier + ") * " + radicalPart + " = " + coeff + " * " + coeff + " * " + sqrtCoefficientMatch[1];
} else {
// — Logic for Binomial Denominator (using conjugate) —
var binomialMatch = den.match(/^(\d+(?:\.\d+)?)\s*\+\s*sqrt\((\d+(?:\.\d+)?)\)$/);
var binomialMatchSubtract = den.match(/^(\d+(?:\.\d+)?)\s*-\s*sqrt\((\d+(?:\.\d+)?)\)$/);
var sqrtFirstBinomialMatch = den.match(/^sqrt\((\d+(?:\.\d+)?)\)\s*\+\s*(\d+(?:\.\d+)?)$/);
var sqrtFirstBinomialMatchSubtract = den.match(/^sqrt\((\d+(?:\.\d+)?)\)\s*-\s*(\d+(?:\.\d+)?)$/);
if (binomialMatch) { // Format: c + sqrt(d)
var c = parseFloat(binomialMatch[1]);
var d = parseFloat(binomialMatch[2]);
var conjugate = c + " – sqrt(" + d + ")";
rationalizedNum = "(" + num + ") * (" + conjugate + ")";
rationalizedDen = "(" + den + ") * (" + conjugate + ") = " + c + "^2 – (sqrt(" + d + "))^2 = " + (c * c) + " – " + d + " = " + (c * c – d);
} else if (binomialMatchSubtract) { // Format: c – sqrt(d)
var c = parseFloat(binomialMatchSubtract[1]);
var d = parseFloat(binomialMatchSubtract[2]);
var conjugate = c + " + sqrt(" + d + ")";
rationalizedNum = "(" + num + ") * (" + conjugate + ")";
rationalizedDen = "(" + den + ") * (" + conjugate + ") = " + c + "^2 – (sqrt(" + d + "))^2 = " + (c * c) + " – " + d + " = " + (c * c – d);
} else if (sqrtFirstBinomialMatch) { // Format: sqrt(d) + c
var d = parseFloat(sqrtFirstBinomialMatch[1]);
var c = parseFloat(sqrtFirstBinomialMatch[2]);
var conjugate = "sqrt(" + d + ") – " + c;
rationalizedNum = "(" + num + ") * (" + conjugate + ")";
rationalizedDen = "(" + den + ") * (" + conjugate + ") = (sqrt(" + d + "))^2 – " + c + "^2 = " + d + " – " + (c * c) + " = " + (d – c * c);
} else if (sqrtFirstBinomialMatchSubtract) { // Format: sqrt(d) – c
var d = parseFloat(sqrtFirstBinomialMatchSubtract[1]);
var c = parseFloat(sqrtFirstBinomialMatchSubtract[2]);
var conjugate = "sqrt(" + d + ") + " + c;
rationalizedNum = "(" + num + ") * (" + conjugate + ")";
rationalizedDen = "(" + den + ") * (" + conjugate + ") = (sqrt(" + d + "))^2 – " + c + "^2 = " + d + " – " + (c * c) + " = " + (d – c * c);
}
else {
// Fallback for unrecognized formats or non-radical denominators
var isNumber = !isNaN(parseFloat(den)) && isFinite(den);
if (isNumber && parseFloat(den) !== 0) {
rationalizedNum = num;
rationalizedDen = den;
} else {
outputElement.textContent = "Unsupported format or division by zero.";
return;
}
}
}
// Display the result
if (rationalizedDen.includes("=")) {
var denParts = rationalizedDen.split("=");
rationalizedDen = denParts[1].trim();
}
outputElement.innerHTML = "Numerator: " + rationalizedNum + " / Denominator: " + rationalizedDen;
}