Tidying messy roots

Mathematics · Number Representation & Place Value · ages 15-16

Name ______________________ Date ____________

To simplify √108 in one square-factor step, which is the largest square factor of 108?
- 4
- 9
- 36
- 54
What is root 50 in simplest surd form?
- 25 root 2
- 5 root 2
- 2 root 5
Root 2 plus root 3 makes root 5.
Circle one: True False
√72 = a × √2, where a is a whole number. What is a?
Answer: ______________
What is 6 over root 2 with a rational denominator?
- 6 root 2
- root 3
- 3 root 2
Which factor equals 1 and makes the denominator of 5/(2√3) rational in one multiplication?
- √6/√6
- √2/√2
- 3/3
- √3/√3
What is √12 + √27 in simplest surd form?
- 5√3
- 5√6
- √39
- 6√3
What is root 2 times (root 8 + root 2)?
Answer: ______________
Which sum has matching square-root parts after simplification, so its coefficients can be added?
- 4 root 2 + 3 root 3
- 3 root 7 + 4 root 7
- 6 root 5 + 2 root 11
Write 10/√5 as a × √5. What is the whole-number coefficient a?
Answer: ______________

Answer key

For grown-ups. Fold this page away before handing over the rest.

36 · 108 = 36 × 3 and 36 = 6². No larger square number divides 108.
5 root 2 · 50 = 25 times 2, and root 25 = 5, so root 50 = 5 root 2.
False · Only matching roots merge. Root 2 plus root 3 cannot combine.
6 · 72 = 36 times 2, and root 36 = 6, so a = 6.
3 root 2 · Multiply top and bottom by root 2: 6 root 2 over 2 = 3 root 2.
√3/√3 · Multiplying by √3/√3 gives denominator 2 × √3 × √3 = 6, so the fraction becomes 5√3/6.
5√3 · √12 = 2√3 and √27 = 3√3. Their root parts now match, so the sum is 5√3.
6 · Root 2 times root 8 is 4, and root 2 times root 2 is 2, giving 6.
3 root 7 + 4 root 7 · Only matching roots merge: 3 root 7 + 4 root 7 = 7 root 7.
2 · Multiply top and bottom by root 5: 10 root 5 over 5 = 2 root 5.