Rationalize Denominator Calculator
Enter your expression to rationalize the denominator.
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A Complete Guide to Rationalize Denominator Calculator
What Does “Rationalize the Denominator” Mean?
In mathematics, “rationalizing the denominator” is a standard process used to rewrite a fraction that contains a radical (like a square root) in its denominator. The goal is to convert the denominator from an irrational number into a rational number (typically an integer).
An irrational number is a number that cannot be expressed as a simple fraction, such as $\sqrt{2}$ or $\pi$. Their decimal representations go on forever without repeating. A rational number is any number that can be expressed as a fraction, such as 5 (which is $\frac{5}{1}$), $\frac{1}{2}$, or 0.75 (which is $\frac{3}{4}$).
Why do we do this?
Mathematical Convention: It is a long-standing rule in algebra to present final answers in their “simplest form.” A fraction with an irrational denominator is generally not considered fully simplified.
Easier Calculations: Before calculators, it was much, much harder to divide a number by an irrational number (like $5 \div \sqrt{2}$) than to divide by an integer ($5\sqrt{2} \div 2$). While modern calculators handle this easily, the convention remains critical for further algebraic manipulation.
This process does not change the actual value of the fraction, only its appearance. We accomplish this by multiplying the fraction by a special form of the number 1.
The Key Concept: Multiplying by a “Special Form of 1”
The most important rule in this process is that you can multiply any number by 1 without changing its value. We will use this rule by creating a fraction that is equal to 1, designed specifically to eliminate the radical in the denominator.
The “special form” we use depends on whether the denominator has one term (a monomial) or two terms (a binomial).
Case 1: Monomial Denominators (e.g., $\sqrt{b}$)
If the denominator is a single term with a square root, like $\frac{a}{\sqrt{b}}$, our goal is to turn $\sqrt{b}$ into $b$.
We know that $\sqrt{b} \times \sqrt{b} = b$.
So, the “special form of 1” we use is $\frac{\sqrt{b}}{\sqrt{b}}$.
When we multiply $\frac{a}{\sqrt{b}} \times \frac{\sqrt{b}}{\sqrt{b}}$, we get $\frac{a\sqrt{b}}{b}$. The denominator is now rational.
Case 2: Binomial Denominators (e.g., $b + \sqrt{c}$)
If the denominator has two terms, like $\frac{a}{b + \sqrt{c}}$, multiplying by just $\frac{\sqrt{c}}{\sqrt{c}}$ won’t work. $(b + \sqrt{c}) \times \sqrt{c} = b\sqrt{c} + c$. You still have a square root!
The trick here is to use the conjugate. The conjugate of a two-term expression is the same expression but with the opposite sign in the middle. This works because of the “Difference of Squares” formula: $(x + y)(x – y) = x^2 – y^2$.
When you multiply a binomial by its conjugate, the “middle” terms (which often contain the square roots) cancel out.
Here is a table of binomials and their conjugates:
Expression | Conjugate | Result of Multiplying Them |
|---|---|---|
$b + \sqrt{c}$ | $b – \sqrt{c}$ | $(b)^2 – (\sqrt{c})^2 = b^2 – c$ |
$b – \sqrt{c}$ | $b + \sqrt{c}$ | $(b)^2 – (\sqrt{c})^2 = b^2 – c$ |
$\sqrt{b} + \sqrt{c}$ | $\sqrt{b} – \sqrt{c}$ | $(\sqrt{b})^2 – (\sqrt{c})^2 = b – c$ |
$\sqrt{b} – \sqrt{c}$ | $\sqrt{b} + \sqrt{c}$ | $(\sqrt{b})^2 – (\sqrt{c})^2 = b – c$ |
In all of these cases, the result is a rational number, and the square roots are gone.
How to Rationalize: Step-by-Step Examples
Our calculator handles all the common forms of this problem. Here is a manual, step-by-step breakdown for each one.
Type 1: Monomial Denominator $\frac{a}{\sqrt{b}}$
This is the simplest case. You just need to multiply the top and bottom by the radical in the denominator.
Problem: Rationalize the fraction $\frac{5}{\sqrt{3}}$
Step 1: Identify the radical in the denominator: $\sqrt{3}$.
Step 2: Create the “special form of 1” using this radical: $\frac{\sqrt{3}}{\sqrt{3}}$.
Step 3: Multiply your original fraction by this new fraction.
$$\frac{5}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}$$Step 4: Multiply the numerators (top) and denominators (bottom) straight across.
Numerator: $5 \times \sqrt{3} = 5\sqrt{3}$
Denominator: $\sqrt{3} \times \sqrt{3} = 3$
Step 5: Write the new, rationalized fraction.
$$\frac{5\sqrt{3}}{3}$$Final Check: Can the integers be simplified? In this case, $\frac{5}{3}$ cannot be simplified, so this is the final answer.
Example with Simplification: Rationalize $\frac{12}{\sqrt{6}}$
Steps 1-4: $\frac{12}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{12\sqrt{6}}{6}$
Step 5 (Simplify): Notice the integers $\frac{12}{6}$. This simplifies to 2.
Final Answer: $2\sqrt{6}$
Type 2: Binomial Denominator $\frac{a}{b + \sqrt{c}}$
Here we must use the conjugate to eliminate the radical.
Problem: Rationalize the fraction $\frac{4}{3 + \sqrt{7}}$
Step 1: Identify the denominator: $3 + \sqrt{7}$.
Step 2: Find its conjugate: $3 – \sqrt{7}$.
Step 3: Create the “special form of 1”: $\frac{3 – \sqrt{7}}{3 – \sqrt{7}}$.
Step 4: Multiply the original fraction.
$$\frac{4}{3 + \sqrt{7}} \times \frac{3 – \sqrt{7}}{3 – \sqrt{7}}$$Step 5: Multiply the numerators. You must use the distributive property.
$4 \times (3 – \sqrt{7}) = (4 \times 3) – (4 \times \sqrt{7}) = 12 – 4\sqrt{7}$
Step 6: Multiply the denominators. Use the Difference of Squares formula $(x+y)(x-y) = x^2 – y^2$.
$(3 + \sqrt{7})(3 – \sqrt{7}) = (3)^2 – (\sqrt{7})^2 = 9 – 7 = 2$
Step 7: Write the new fraction.
$$\frac{12 – 4\sqrt{7}}{2}$$Step 8 (Simplify): You must check if all three terms ($12$, $-4$, and $2$) can be divided by a common number. In this case, all three are divisible by 2.
$\frac{12}{2} – \frac{4\sqrt{7}}{2} = 6 – 2\sqrt{7}$
Final Answer: $6 – 2\sqrt{7}$
Type 3: Binomial Denominator $\frac{a}{b – \sqrt{c}}$
This is identical to Type 2, but the conjugate will have a + sign.
Problem: Rationalize $\frac{10}{6 – \sqrt{11}}$
Step 1: Denominator: $6 – \sqrt{11}$
Step 2: Conjugate: $6 + \sqrt{11}$
Step 3: Form of 1: $\frac{6 + \sqrt{11}}{6 + \sqrt{11}}$
Step 4: Multiply: $\frac{10}{6 – \sqrt{11}} \times \frac{6 + \sqrt{11}}{6 + \sqrt{11}}$
Step 5 (Numerator): $10 \times (6 + \sqrt{11}) = 60 + 10\sqrt{11}$
Step 6 (Denominator): $(6)^2 – (\sqrt{11})^2 = 36 – 11 = 25$
Step 7 (New Fraction): $\frac{60 + 10\sqrt{11}}{25}$
Step 8 (Simplify): All three terms ($60$, $10$, and $25$) are divisible by 5.
$\frac{60 \div 5}{25 \div 5} + \frac{10\sqrt{11} \div 5}{25 \div 5} = \frac{12 + 2\sqrt{11}}{5}$
Final Answer: $\frac{12 + 2\sqrt{11}}{5}$
Type 4: Binomial Denominator $\frac{a}{\sqrt{b} + \sqrt{c}}$
This type looks more complex, but the process is the same.
Problem: Rationalize $\frac{5}{\sqrt{6} + \sqrt{3}}$
Step 1: Denominator: $\sqrt{6} + \sqrt{3}$
Step 2: Conjugate: $\sqrt{6} – \sqrt{3}$
Step 3: Form of 1: $\frac{\sqrt{6} – \sqrt{3}}{\sqrt{6} – \sqrt{3}}$
Step 4: Multiply: $\frac{5}{\sqrt{6} + \sqrt{3}} \times \frac{\sqrt{6} – \sqrt{3}}{\sqrt{6} – \sqrt{3}}$
Step 5 (Numerator): $5 \times (\sqrt{6} – \sqrt{3}) = 5\sqrt{6} – 5\sqrt{3}$
Step 6 (Denominator): $(\sqrt{6})^2 – (\sqrt{3})^2 = 6 – 3 = 3$
Step 7 (New Fraction): $\frac{5\sqrt{6} – 5\sqrt{3}}{3}$
Step 8 (Simplify): The integers are $5$, $-5$, and $3$. They do not share a common divisor (other than 1).
Final Answer: $\frac{5\sqrt{6} – 5\sqrt{3}}{3}$
Type 5: Binomial Denominator $\frac{a}{\sqrt{b} – \sqrt{c}}$
This is the final case, and it follows the exact same pattern.
Problem: Rationalize $\frac{18}{\sqrt{10} – \sqrt{1}}$ (Note: $\sqrt{1} = 1$)
Step 1: Denominator: $\sqrt{10} – 1$
Step 2: Conjugate: $\sqrt{10} + 1$
Step 3: Form of 1: $\frac{\sqrt{10} + 1}{\sqrt{10} + 1}$
Step 4: Multiply: $\frac{18}{\sqrt{10} – 1} \times \frac{\sqrt{10} + 1}{\sqrt{10} + 1}$
Step 5 (Numerator): $18 \times (\sqrt{10} + 1) = 18\sqrt{10} + 18$
Step 6 (Denominator): $(\sqrt{10})^2 – (1)^2 = 10 – 1 = 9$
Step 7 (New Fraction): $\frac{18\sqrt{10} + 18}{9}$
Step 8 (Simplify): All three terms ($18$, $18$, and $9$) are divisible by 9.
$\frac{18\sqrt{10}}{9} + \frac{18}{9} = 2\sqrt{10} + 2$
Final Answer: $2\sqrt{10} + 2$
How to Use Our Rationalize Denominator Calculator
While the manual process is essential for learning, our calculator makes it fast, accurate, and educational. It’s designed to help you solve problems and understand the solution.
Step 1: Select Your Expression Type
Look at your problem and match it to one of the options in the dropdown menu.
a / sqrt(b)a / (b + sqrt(c))a / (b - sqrt(c))a / (sqrt(b) + sqrt(c))a / (sqrt(b) - sqrt(c))
Step 2: Enter Your Numbers
Input the values from your problem into the corresponding a, b, and c fields.
ais always the numerator.bandcare the numbers in the denominator.
Step 3: Click “Calculate”
The calculator instantly performs all the steps described above. It will:
Find the correct conjugate.
Multiply the numerator and denominator.
Simplify the resulting expression by finding the Greatest Common Divisor (GCD) for all terms.
Display the final, fully simplified answer.
The calculator also shows the step-by-step math it used, so you can check your own work or learn the process.
Advanced Feature: 🤖 AI-Powered Explanations
Sometimes, just seeing the steps isn’t enough. You might wonder why a certain step was taken. We’ve integrated an advanced AI feature to help.
After you get a solution, an “Explain Solution” button will appear.
When you click it, our AI acts as a personal math tutor. It analyzes your exact problem and the final solution and then generates a simple, conversational explanation of the entire process from start to finish.
This feature turns the calculator from a simple answer-finder into a powerful learning tool. It’s perfect for when you’re stuck on homework and need a clear, plain-English explanation of how to get the right answer.
Frequently Asked Questions (FAQs)
Q: What is a “conjugate” again?
A: A conjugate is a binomial formed by flipping the sign between two terms. The conjugate of $(x + y)$ is $(x – y)$. We use it because multiplying a binomial by its conjugate results in a “difference of squares” ($x^2 – y^2$), which removes the square roots.
Q: Why do we have to rationalize denominators?
A: It is a mathematical convention to write expressions in their simplest form. A fraction with a radical in the denominator is not considered simple. It also makes it much easier to perform further calculations with the expression.
Q: What is the most common mistake?
A: The most common mistake is forgetting to multiply the numerator by the conjugate. Whatever you do to the denominator, you must do the exact same thing to the numerator, or you change the fraction’s value. Another common error is in simplification: to simplify $\frac{a + b}{c}$, both $a$ and $b$ must be divisible by $c$.
Q: Does this calculator handle cube roots?
A: This calculator is designed specifically for square roots. Rationalizing cube roots (like $\frac{1}{\sqrt[3]{2}}$) is a different process that involves using the “sum or difference of cubes” formulas, not the “difference of squares.”
Q: What happens if the denominator becomes zero?
A: Our calculator is built to check for this. For an expression like $\frac{a}{\sqrt{b} – \sqrt{c}}$, if $b = c$, the denominator is $0$, which is undefined. The calculator will show an error message.
