Simplifying and Combining Radicals

Pull out the perfect squares, then collect like terms.

12 problems · about 21 minutes · N-RN.A.2, A-SSE.A.2

What this lesson teaches

The student simplifies radical expressions and adds, multiplies and rationalises them.

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.

Straightforward practice. Get the method working first.

5 problems

√45 = a√5. What is a?
Answer 3
Why 3.
Why can √2 + √3 not be combined?
Answer They are unlike radicals, the way 2x and 3y are unlike terms.
Why Different radicands make them unlike terms.
√18 = a√2. What is a?
Answer 3
Why 18 = 9 × 2.
√75 = a√3. What is a?
Answer 5
Why 75 = 25 × 3.
3√5 + 4√5 = a√5. What is a?
Answer 7
Why Add the coefficients.

The same ideas with more to keep track of.

3 problems

√2 × √8 equals what number?
Answer 4
Why √16.
√(9 + 16) equals what?
Answer 5
Why Add inside first: √25.
6 ÷ √3, rationalised, is a√3. What is a?
Answer 2
Why Multiply top and bottom by √3 to get 6√3 ÷ 3.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Which pairs are like radicals once simplified?
Answer √8 and √18; √27 and √12
Why Simplify each one first and compare what stays under the root.
(2 + √3)(2 − √3) equals what number?
Answer 1
Why A difference of squares: 4 − 3.
The Simplify: √12 simplifies to a√3. What is a?
Answer 2
Why 2, giving 2√3.
The Diagonal: A square has area 32. Its side is a√2. What is a?
Answer 4
Why 4, so the side is 4√2.