Rationalizing Denominators
Remove roots from denominators by multiplying by a conjugate.
Key idea
Remove roots from denominators by multiplying by a conjugate.
In depth
To remove a root from the denominator, multiply top and bottom by something that turns the denominator rational. For a binomial like a + √b, multiply by its conjugate a − √b to use difference of squares.
Worked example
1/√3 = √3/3. And 1/(2+√3) × (2−√3)/(2−√3) = (2−√3)/(4−3) = 2 − √3.
Formula
Real-world uses
- Writing expressions in standard form in exams and engineering
- Making calculations with surds easier
Common mistake
Multiplying only the denominator, which changes the value.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Rationalizing Denominators?
Remove roots from denominators by multiplying by a conjugate. It is a topic at the advanced level in Arithmetic & Number Sense.
What is the formula for Rationalizing Denominators?
The key formula is 1/√a = √a/a.
Where is Rationalizing Denominators used in real life?
Real-world examples: Writing expressions in standard form in exams and engineering; Making calculations with surds easier.
What should I learn before Rationalizing Denominators?
Make sure you are comfortable with Square Roots of Non-squares, the previous topic in Arithmetic & Number Sense.
What is a common mistake with Rationalizing Denominators?
Multiplying only the denominator, which changes the value.
How can I practise Rationalizing Denominators?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Rationalizing Denominators?
Move on to Laws of Exponents next.
Related advanced topics in Arithmetic & Number Sense
- Repeating Decimals – Decimals with a repeating block are rational and convertible to fractions.
- Compound Interest – Interest earned on principal and past interest.
- Inverse Proportion – When one quantity doubles, the other halves.
- Rational Numbers – Numbers expressible as p/q with integers p, q, q≠0.
- Irrational Numbers – Real numbers that cannot be written as a fraction, such as √2 and π.
- Real Numbers – All rational and irrational numbers; the continuous number line.