A root splits across multiplication: √48 = √16 × √3 = 4√3. Pull out the largest perfect square you can find.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
It does not split across addition
√(9 + 16) is 5, not 3 + 4. Roots distribute over products and quotients only.
Like radicals
Only matching radicals combine: 2√3 + 5√3 = 7√3. But √2 + √3 stays as it is, the same way 2x + 3y does.
Simplify first
Two radicals may look unlike until simplified. √8 + √18 becomes 2√2 + 3√2, which is 5√2.
Rationalising
A radical in a denominator is traditionally cleared by multiplying top and bottom by that radical.
Two-term denominators
For something like 1 ÷ (2 + √3), multiply by the conjugate 2 − √3. The difference of squares removes the root.
Pull out the perfect squares
√72 = √(36 × 2) = 6√2. Find the largest perfect square factor and take its root outside. Working with the largest factor gets you to simplest form in one step rather than several.
Radicals combine only when alike
3√2 + 5√2 = 8√2, but 3√2 + 5√3 cannot be combined. The radical part must match exactly, which is the same rule as for like terms in polynomials — and simplifying first often reveals matches that were hidden.
Multiplying is easier than adding
√a × √b = √(ab) for non-negative a and b. Multiplication combines freely; addition does not. This asymmetry surprises people, and assuming √(a + b) = √a + √b is a frequent and serious error.
Clearing a radical from the denominator
Multiply top and bottom by the radical, or by the conjugate if the denominator is a sum. The convention predates calculators, when dividing by a decimal was harder than multiplying, but it remains standard because it makes forms comparable.
Step 2: Try It Yourself
Tap and try it out.
- Point(4, 2)
Step 3: Watch an Example
One step at a time.
Watch Amara Simplify √50 + √8
These two look unlike until they are simplified.
- Step 1
She splits 50 as 25 × 2, so √50 = 5√2.
Step 4: Your Turn
Practice makes it stick.
The Simplify
Problem 1 of 2
√12 simplifies to a√3. What is a?
The Diagonal
Problem 2 of 2
A square has area 32. Its side is a√2. What is a?
Simplest Form
1 of 8
√18 = a√2. What is a?
2 of 8
√75 = a√3. What is a?
3 of 8
3√5 + 4√5 = a√5. What is a?
4 of 8
√2 × √8 equals what number?
5 of 8
√(9 + 16) equals what?
6 of 8
6 ÷ √3, rationalised, is a√3. What is a?
7 of 8
Which pairs are like radicals once simplified?
8 of 8
(2 + √3)(2 − √3) equals what number?
Step 5: Quick Check
Show what you know.
Question 1 of 2
√45 = a√5. What is a?
Question 2 of 2
Why can √2 + √3 not be combined?
What You Learned
- Split a root across products, never across sums.
- Only like radicals combine, so simplify before deciding.
- Clear a radical from a denominator with the radical itself, or with a conjugate.