Radicals
Simplify square roots and rationalize denominators.
Objective
Simplify square roots and rationalize denominators.
What it means
A radical is simplified when the number under the root has no perfect-square factor left, and when there is no radical left in the denominator of a fraction.
Key terms
- Radicand
- The number under the radical sign
- Perfect square
- A number that is an integer squared, such as 4, 9, 16, 25
- Rationalizing
- Removing a radical from the denominator by multiplying top and bottom by that radical
Rules to remember
- sqrt(a*b) = sqrt(a) * sqrt(b)
- Pull out the largest perfect-square factor of the radicand.
- To rationalize 1/sqrt(n), multiply numerator and denominator by sqrt(n).
Step by step
- 1Find the largest perfect-square factorBreak the radicand into (perfect square) * (leftover).
- 2Split the radicalTake the square root of the perfect-square part.
- 3Rationalize if neededMultiply by the radical over itself to clear the denominator.
Common mistakes
- Leaving a radical in the denominator.
- Missing a larger perfect-square factor and stopping too early.
Tips
- Check 4, 9, 16, 25, 36, 49, 64 as possible factors first.
- A fully simplified radicand has no perfect-square factors other than 1.