The product and quotient rules for radicals, simplifying square roots and higher roots, radicals with different indices, powers of radicals and roots of roots.
In this lesson, variables under a radical stand for positive numbers.
The product and quotient rules
Simplest radical form
A square root is in simplest form when the radicand has no perfect-square factor other than \(1\). The product rule, read backwards, lets you pull those factors out.
For variables there's a quick rule: divide the exponent by the index. The quotient is the exponent outside; the remainder stays inside. For \(\sqrt{x^7}\):\(7 \div 2 = 3\) remainder \(1\), so \(x^3\sqrt{x}\).
Different indices
To multiply \(\sqrt{3}\) by \(\sqrt[3]{2}\), first rewrite both with a common index, the LCM of the indices.