Definitions:

From Wolfram MathWorld:

Any nonnegative real number x has a unique nonnegative square root r; this is called the principal square root and is written $$r=x^{(\frac{1}{2})}$$ or $$r = \sqrt{x}$$ In common usage, unless otherwise specified, "the" square root is generally taken to mean the principal square root. Where √ is called the radical sign or radix.

Square Root -- from Wolfram MathWorld

Properties of square roots and radicals guide us on how to deal with roots when they appear in algebra.

For all positive real numbers x and y, $$\sqrt{x}  * \sqrt{y} = \sqrt{xy}$$ $$\sqrt{x + y} \ne \sqrt{x}  + \sqrt{y}$$ For all positive real numbers x and y, y ≠ 0: $$\sqrt{\frac{x}{y}} = \frac{\sqrt{x}}{\sqrt{y}}$$ $$a\sqrt{x}   + b\sqrt{x} = {(a + b)}\sqrt{x}$$ $$a\sqrt{x}   - b\sqrt{x} = {(a - b)}\sqrt{x}$$ $$a\sqrt{x}   = \sqrt{a^2 * x}$$ $$\sqrt[n]{x} = a, \space\space if \space\space n \space\space is \space\space odd$$ $$\sqrt[n]{x} = |a|, \space\space if \space\space n \space\space is \space\space even$$

A general example to help you recognize patterns and spot the information you're looking for

$$\sqrt{y^4} = y^2$$ $${(4x^2)^{\frac{1}{2}}} = \pm{(\sqrt{4}   * \sqrt{x^2}) = \pm2x}$$ $$\sqrt{(3x^2 + 6x^2)} = \sqrt{9x^2} = 3x$$ $$\sqrt{80} = \sqrt{(16 * 5)} = \sqrt{16}   * \sqrt{5} = 4\sqrt{5}$$ $$13\sqrt{a} - 4\sqrt{a} = {(13 - 4)}\sqrt{a} = 9\sqrt{a}$$ Careful!!! $$\sqrt{a + b} \ne \sqrt{a} + \sqrt{b}$$ $$\sqrt{a - b} \ne \sqrt{a} - \sqrt{b}$$ $$\sqrt{a^2 + b^2} \ne a + b$$

It’s used in construction, like finding amount of materials for construction, etc.

Understanding Square Roots

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Length: 1:21