Product Property of Radicals
Simplify radical expressions using the product property of radicals in Grade 9 Algebra. Split or combine radicals by factoring out perfect squares under the radical sign.
Key Concepts
Property The square root of a product equals the product of the square roots of the factors. $$ \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \text{ where } a \ge 0 \text{ and } b \ge 0 $$ Explanation Think of this as a matchmaking rule! Numbers outside the radicals multiply together, and numbers inside the radicals (radicands) multiply together. You can't mix and match an outsider with an insider. Keep them in their own groups, combine them, and then simplify the result to get your final, perfect couple. Examples $ \sqrt{12}\sqrt{3} = \sqrt{36} = 6 $ $ 5\sqrt{3} \cdot 2\sqrt{7} = (5 \cdot 2)(\sqrt{3} \cdot \sqrt{7}) = 10\sqrt{21} $ $ (4\sqrt{5})^2 = 4^2 \cdot (\sqrt{5})^2 = 16 \cdot 5 = 80 $.
Common Questions
What is the product property of radicals?
The product property states that √(a·b) = √a · √b for non-negative a and b. You can split a radical over multiplication or combine two radicals with the same index into one. Use it to simplify by pulling out perfect square factors.
How do you simplify a radical expression using the product property?
Factor the radicand to identify perfect square factors. Apply the product property to separate the perfect square under its own radical, then evaluate it. Multiply the whole number result by the remaining radical.
What is the most common strategy for simplifying √72?
Factor 72 as 36 · 2, where 36 is a perfect square. Apply the product property: √72 = √36 · √2 = 6√2. Always look for the largest perfect square factor to reach the simplest form in one step.