The Product Rule for Radicals
Apply the product rule for radicals in Grade 10 algebra: √(a·b) = √a · √b to simplify radical expressions by factoring out perfect squares from under the radical sign.
Key Concepts
Given that $a$ and $b$ are real numbers and $n$ is an integer greater than 1, $$\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$$.
$\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25}\sqrt{2} = 5\sqrt{2}$ $\sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36}\sqrt{2} = 6\sqrt{2}$ $\sqrt[3]{128} = \sqrt[3]{64 \cdot 2} = \sqrt[3]{64}\sqrt[3]{2} = 4\sqrt[3]{2}$.
This rule is your secret weapon! It lets you break a big, scary number under a root into a product of smaller, manageable ones. Your mission is to find a perfect square factor hiding inside, pull its root out to the front, and leave the non perfect leftover part inside. This makes simplifying complex radicals so much easier.
Common Questions
What is the product rule for radicals?
√(a·b) = √a · √b, where a≥0 and b≥0. This allows you to split a radical into simpler parts or combine two radicals into one.
How do you simplify √72 using the product rule?
Factor 72 as 36×2: √72 = √(36×2) = √36 · √2 = 6√2.
How do you multiply √3 · √12?
Apply the product rule: √3 · √12 = √(3×12) = √36 = 6.