Multiplying and Dividing Square Roots
Multiplying square roots uses √a × √b = √(ab), so you multiply the numbers under one root sign. Dividing square roots works the same way using √a ÷ √b = √(a/b). Both rules let you simplify square roots in a calculation.
Video Lesson
Watch and learn the basics
🎬 Did this video explain it clearly?
Flashcards
Review key concepts visually
%20Multiplying%20and%20Dividing%20Roots.webp)
%20Example%20Multiplication.webp)
%20Example%20Division.webp)
Square Root Rules: Multiply and Divide
- To multiply square roots, .
- To divide square roots, .
Multiplying Square Roots
- Example 1: .
- Example 2: .
Dividing Square Roots
- Example 1: .
- Example 2: .
Practice Questions
Test your understanding
Simplify .
Correct! 🎉 +10 pointsNot quite right
We use the multiplication rule for square roots: . We break 50 into , so .
Simplify .
Correct! 🎉 +10 pointsNot quite right
We use the multiplication rule for square roots: . We break 72 into , so .
Simplify .
Correct! 🎉 +20 pointsNot quite right
We use the division rule for square roots, which states . So, .
Simplify .
Correct! 🎉 +20 pointsNot quite right
We use the division rule for square roots, which states . So, .
Simplify .
Correct! 🎉 +20 pointsNot quite right
We use the division rule for square roots, which states . So, .
What is ?
Correct! 🎉 +30 pointsNot quite right
We use the rule . So, . Now rewrite as , simplifying as 3, we get the result .
Want to see the full working?
Interactive Activity
Practice multiplying and dividing square roots
Loading interactive widget...
Students Also Ask
The questions students bump into most on this topic
The rule for multiplying surds is √a × √b = √(ab), where a and b are non-negative. You multiply the numbers under one square root sign, then simplify if a perfect square factor appears. The rule also works in reverse, letting you split one root into two.
The rule for dividing surds is √(a/b) = √a ÷ √b, where b is not zero. You divide the numbers under one square root sign, or split the root of a fraction into a fraction of roots. The denominator must never be zero.
Yes. You can use both surd rules from left to right or from right to left. Left to right splits one square root into two, which helps you simplify. Right to left combines two square roots into one, which often makes a calculation much easier.
Yes. When a surd is written as a fraction instead of a division, the rule works in exactly the same way. The square root of a fraction equals the square root of the top divided by the square root of the bottom, as long as the bottom is not zero.
The denominator cannot be zero because you cannot divide by zero. The quotient rule √(a/b) = √a ÷ √b needs b to be a positive number so the fraction and its square root both have a value. Any non-zero denominator is fine.