Practice · Grade 9 radicals
Dividing Radicals — Practice
Practice dividing square roots with the quotient rule, simplifying, and rationalizing — increasing difficulty, each with a hint and worked solution. Free.
Q1 of 6
0 correct
Use the quotient rule to compute √50 ÷ √2.
√50 ÷ √2
Quick answer
What's the best way to practice dividing radicals?
For every problem, apply the quotient rule first: √a ÷ √b = √(a ÷ b). Then check whether the quotient is a perfect square (whole-number result), whether you can pull out perfect squares (√18 = 3√2), or whether a root is left in the denominator. If a root remains in the denominator, rationalize it with the denominator's root: √a/√b = √(a · b)/b, and reduce the fraction at the end.
HowTo
A 4-step solving strategy
This order fits any problem of the form √a ÷ √b with a ≥ 0 and b > 0.
- 1Step 1 of 4
Apply the quotient rule
Combine the two roots into one: √a ÷ √b = √(a ÷ b). √50 ÷ √2 becomes √(50 ÷ 2). Important: the radicands get divided, not subtracted.
- 2Step 2 of 4
Work out the quotient under the root
Divide the numbers under the root, e.g. 50 ÷ 2 = 25. If the result is a perfect square (4, 9, 16, 25 …), take the root exactly: √25 = 5.
- 3Step 3 of 4
Simplify — pull out perfect squares
No clean root? Look for square factors: √80 = √(16 · 5) = 4√5. A root like √3 is already simplest because 3 has no square factor.
- 4Step 4 of 4
Rationalize the denominator and reduce
If a root remains in the denominator, multiply by it: √2/√3 = √6/3. Then reduce numerator and denominator if they share a factor: 4√5/8 = √5/2.
Examples
Worked practice examples
Four typical problem types from Grade 9 tests. Try each one yourself first, then compare.
Easy
Clean root: √72 ÷ √2
√72 ÷ √2 = √(72 ÷ 2)
= √36
= 6
Check: 6 · √2 = √36 · √2 = √72 ✓
The quotient 36 is a perfect square, so the result is exactly a whole number.
Medium
Simplified root: √6 ÷ √2
√6 ÷ √2 = √(6 ÷ 2)
= √3
≈ 1.7321
3 has no square factor → √3 is simplest
Not every root comes out clean. √3 stays as an irrational value.
Medium
Rationalizing: √2 ÷ √3
(√2 · √3) ÷ (√3 · √3)
= √6 ÷ 3
≈ 0.8165
Denominator root-free: 3 instead of √3 ✓
Multiplying by the denominator's root makes the denominator rational.
Hard
Boss: fully simplify √10 ÷ √8
(√10 · √8) ÷ 8 = √80 ÷ 8
√80 = 4√5, so 4√5 ÷ 8
= √5 ÷ 2 ≈ 1.1180
Check: (√5/2)² = 5/4 = 1.25 = 10/8 ✓
Three techniques at once: rationalize, pull out the square, reduce.
Pitfalls
Common mistakes — and how to avoid them
These five traps come up again and again when dividing radicals.
Subtracting radicands instead of dividing
√12 ÷ √3 is √(12 ÷ 3) = √4 = 2, not √(12 − 3) = √9. The quotient rule divides the numbers under the roots.
Leaving a root in the denominator
1/√3 or √2/√3 count as unfinished. Multiply by the denominator's root: √2/√3 = √6/3. A rational denominator is the standard form.
Missing perfect squares
√80 isn't "done" — it's 4√5. Always check whether a perfect square (4, 9, 16, 25 …) hides as a factor in the radicand.
Not reducing after rationalizing
4√5/8 isn't simplest yet. The numerator 4 and denominator 8 share a factor of 4: 4√5/8 = √5/2. Always reduce at the end.
Allowing a negative or zero radicand
For real roots a must be ≥ 0, and the denominator b must even be > 0 — you can't divide by √0.
Study
Practice with a plan — three quick tips
Know your perfect squares
If you instantly recognize 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, you can tell in seconds whether a root comes out clean or whether a square needs pulling out.
Solve first, then look at the answer
Write out your working before you reveal the hint. Active recall is three to four times more effective for learning than passive reading.
For every wrong answer, ask why
Was it a missed perfect square, an unreduced fraction, or a root left in the denominator? Note the cause — and next time you'll spot the mistake immediately.
FAQ
Frequently asked questions about practicing
Glossary
Key terms in one sentence
- Radical (root)
- The square root √a is the non-negative number whose square is a.
- Radicand
- The number under the root sign, e.g. the 12 in √12.
- Quotient rule
- √a ÷ √b = √(a ÷ b), valid for a ≥ 0 and b > 0.
- Rationalize
- Make the denominator root-free by multiplying by the denominator's root.
- Perfect square
- A square number like 4, 9, 16, 25 whose root is a whole number.
- Irrational number
- A number like √3 that can't be written as a fraction of two integers.
- Boss question
- The last and hardest problem in a practice set, combining several techniques.