Explanation:
To multiply and divide square roots, we use the following rules:
Multiplying square roots:
To multiply square roots, we can multiply the radicands (the numbers inside the square roots) and simplify the result. For example:
√a * √b = √(a*b)
Dividing square roots:
To divide square roots, we can divide the radicands and simplify the result. For example:
√a / √b = √(a/b)
We can also simplify square roots by factoring out perfect squares from the radicand. For example:
√48 = √(16*3) = √16 * √3 = 4√3
When dividing square roots with larger numbers or variables, we may need to use additional algebraic techniques such as rationalizing the denominator. This involves multiplying the numerator and denominator by a factor that eliminates any radicals in the denominator. For example:
(√a + √b) / (√a - √b) = [(√a + √b) / (√a - √b)] * [(√a + √b) / (√a + √b)] = (a + 2√ab + b) / (a - 2√ab + b)
These rules and techniques are essential for simplifying and manipulating expressions that involve square roots.