Radical Calculator

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Radical Calculator
Calculate N-th Root (Decimal)Simplify Radical Expression
Result:
function calculateRadical(){var coef=parseFloat(document.getElementById('coefficient').value);var n=parseInt(document.getElementById('index').value);var x=parseFloat(document.getElementById('radicand').value);var type=document.getElementById('calc_type').value;if(isNaN(coef))coef=1;if(isNaN(n)||isNaN(x)){alert('Please enter a valid Index and Radicand');return;}if(n===0){alert('Index cannot be zero');return;}if(n%2===0 && x<0){alert('Even roots of negative numbers result in imaginary numbers');return;}var display=document.getElementById('expressionDisplay');var final=document.getElementById('finalResult');var decimal=document.getElementById('decimalResult');var ansDiv=document.getElementById('answer');ansDiv.style.display='block';display.innerHTML=coef+' × &sup'+n+';√('+x+')';var decimalVal=coef*Math.pow(Math.abs(x),1/n);if(x<0)decimalVal=-decimalVal;if(type==='root'){final.innerHTML=decimalVal.toFixed(6);decimal.innerHTML='';}else{var simplifiedCoef=coef;var simplifiedRadicand=x;var isNeg=(x=2;i–){var power=Math.pow(i,n);if(absX%power===0){simplifiedCoef*=i;absX/=power;}}simplifiedRadicand=isNeg?-absX:absX;if(simplifiedRadicand===1){final.innerHTML=simplifiedCoef;}else if(simplifiedRadicand===-1){final.innerHTML='-'+simplifiedCoef;}else{var radicalSign=(n===2)?'√':'&sup'+n+';√';final.innerHTML=(simplifiedCoef===1?":simplifiedCoef=== -1?'-':simplifiedCoef)+radicalSign+'('+simplifiedRadicand+')';}decimal.innerHTML='Decimal Approximation: '+decimalVal.toFixed(6);}}

How to Use the Radical Calculator

Our radical calculator is a versatile tool designed to help students, engineers, and math enthusiasts simplify complex radical expressions and calculate the precise n-th root of any number. Whether you are dealing with basic square roots or complex higher-order roots, this tool provides both the simplified radical form and the decimal equivalent.

To get started, follow these simple steps:

Coefficient (a)
The number multiplying the radical. If you just have a root like √20, the coefficient is 1.
Index (n)
The "degree" of the root. Use 2 for square roots, 3 for cube roots, etc.
Radicand (x)
The value inside the radical symbol that you want to calculate or simplify.

How Radical Simplification Works

Simplifying a radical means reducing the number under the radical to the smallest possible integer by "pulling out" perfect powers. The radical calculator uses the following mathematical property:

&supn;√(an × b) = a × &supn;√b

The logic involves these steps:

  • Identify the index (n) of the radical.
  • Find the largest perfect n-th power that is a factor of the radicand.
  • Take the n-th root of that factor and multiply it by the existing coefficient.
  • Leave the remaining factor under the radical sign.

Examples of Radical Calculations

Example 1: Simplify √72

1. The index (n) is 2 (square root).
2. Factors of 72 include 36, which is a perfect square (62).
3. √72 = √(36 × 2).
4. Pull out the 6: 6√2.
5. Result: 6√2 &approx; 8.485281

Example 2: Simplify Cube Root of 54 (³√54)

1. The index (n) is 3.
2. Perfect cubes: 8, 27, 64…
3. 54 is divisible by 27 (33).
4. ³√54 = ³√(27 × 2) = 3 × ³√2.
5. Result: 3³√2 &approx; 3.779763

Common Questions

Can the radical calculator handle negative numbers?

Yes, but with limitations. If the index is odd (like 3, 5, or 7), the calculator will provide the negative root. If the index is even (like 2 or 4), the root of a negative number is an imaginary number (using 'i'), which most standard calculators treat as an error unless they support complex numbers.

What is the difference between an index and a radicand?

The index tells you how many times a number must be multiplied by itself to reach the radicand. The radicand is the target number sitting inside the radical symbol. For example, in ³√8, the index is 3 and the radicand is 8.

What does it mean to rationalize the denominator?

In mathematics, it is often standard practice to remove radicals from the bottom (denominator) of a fraction. While this specific radical calculator focuses on simplification and roots, the results provided here can be used as a step in that process.

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