Radicals
Radical arithmetic
ALEKS placement
Radicals add like variables — only identical radicands combine.
What this covers
- Simplify every radical first; unlike terms often become like ones.
- Add or subtract the coefficients of matching radicands.
- Multiply by multiplying coefficients and radicands separately.
- Rationalize a denominator by multiplying by the radical or by its conjugate.
Worked example
Worked example
Simplify √50 + 3√2.
- 5√2+3√2√50 = √25·√2 = 5√2, which now matches the other term.
- 8√2Add the coefficients.
Another worked example
Multiply √6 · √15, then simplify.
- √6⋅√15=√6⋅15=√90Both radicands are nonnegative, so the roots multiply under one root: 6 · 15 = 90.
- √90=√9⋅10=√9√10=3√109 is the largest perfect square dividing 90. Its root, 3, comes outside.
A common mistake
A mistake Lemma catches
From 5√2+3√2
5√2+3√2=8√45√2+3√2=8√2
“Like terms add their coefficients and keep what they count: 5√2 + 3√2 = (5 + 3)√2 = 8√2, just as 5x + 3x = 8x. Adding the radicands too gives 8√4 = 16, far more than 5√2 + 3√2 ≈ 11.3.”
Try one
Sample problem
Simplify.
6√7−4√7Practice radical arithmetic free
No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.
Where this fits
Before this