The area of a square is calculated using [math]A=a^2,[/math] the formula where [i]a[/i] is the length of the side. The square root is used as a inverse operation to squaring.[br][br][color=#0000ff]Let [math]\large \textcolor{blue}{x\geq 0.}[/math] The [i][b]square root of [/b][/i][b]x[/b] means the real number, for which[/color][br] [br]  [math]\large \textcolor{blue}{\sqrt x \geq 0 \;\;\text{ and }\;\; (\sqrt x)^2=x.}[/math][br] [br][br]Taking the square root can be done separately, if all conditions are satisfied:[br][br]  1. [math]\large \textcolor{blue}{\sqrt{ab}=\sqrt a \sqrt b,\;\;\;a,b \geq 0}[/math][br][br]  2. [math]\large \textcolor{blue}{\sqrt\frac a b=\frac \sqrt a \sqrt b,\;\;\;\;\;\;\; a\geq 0, \; b>0 }[/math][br] [br]Both formulas require that the factors of product or division are positive. This is easy to check when we calculate with numbers. If our calculation includes unknown values, i.e. parameters, these conditions must be verified separately. Examples 6 through 8 deal with these situations.[br][br][u]If you are not sure about the positivity, do not take square roots separately![/u][br][br]Examples:[br][br] 1.  [math] \sqrt 4=\sqrt{2^2}=2[/math][br] [br] 2.  [math] \sqrt 4=\sqrt{(-2)^2}=|-2|=2 [/math][br] [br] 3.  [math] \sqrt{16+9}=\sqrt{25}=5[/math][br] [br] 4.  [math] \sqrt{2^2+6^2}=\sqrt{4+36}=\sqrt{4\cdot 10}=\sqrt 4\cdot \sqrt{10}=2\sqrt{10}[/math][br] [br] 5.  [math]\sqrt\frac 4 9=\frac \sqrt 4 \sqrt 9=\frac 2 3 [/math][br] [br] 6. [math]\sqrt x^2=|x|, [/math] because the value of [i]the variable x[/i] can be any real number. However, by definition, the value of the square root must be at least zero. Absolute values guarantee this.[br][br][br]

[color=#0000ff]Example 6[/color]. [math] \sqrt{16ab^2}=\sqrt{16}\sqrt a\sqrt {b^2}=4\sqrt a|b| [/math][br][br]Inside the square root is multiplication. The factors of this multiplication are 16, [i]a[/i] and [i]b[sup]2[/sup][/i]. Of these factors, 16 is a positive number. Parameter [i]b[/i] can be any number, because it is always at least 0 when raised to another. Only factor [i]a[/i] can make this result negative, i.e. parameter [i]a[/i] must be positive to begin with. Since parameter [i]b[/i] can be either positive or negative, absolute value signs must be placed around it after taking the square root. In this case, we can be sure that the result is positive by definition. [br][br][br][color=#0000ff]Example 7. [/color] [math] \sqrt \frac{9a^3b}{b^3c^4}=\sqrt\frac{9a^3}{b^2c^4}=\frac{\sqrt 9\sqrt{a^2\cdot a}}{\sqrt {b^2}\sqrt{c^4}}=\frac{3a\sqrt a}{|b|c^2} [/math][br][br]In this example, the factors are 9, [i]a[/i], [i]b[/i], and[i] c. [/i]After simplifying, the numerator has [math]9a^3[/math] and the denominator [math]b^2c^4.[/math] Since the power of both terms of the denominator is an even number, we know that the result is positive. In the numerator, the power of parameter [i]a [/i]is an odd number, i.e. the result is the same as the value of [i]parameter a[/i]. Therefore, parameter [i]a [/i] must be a positive number from the start. [br][br]Factor [math]\sqrt{c^4}=\sqrt{(c^2)^2}=c^2.[/math] Therefore, [i]c[/i] can be any number and the square root is calculable. We must claim the value of the parameter [i]b[/i] to be positive by means of absolute values. [br][br][br][color=#0000ff]Example 8. [/color] [math] \sqrt{ab}\left (\sqrt\frac a b+\sqrt \frac b a\right )=\sqrt{ab}\sqrt\frac a b+\sqrt{ab}\sqrt\frac b a =\sqrt \frac{ab\cdot a} b+\sqrt \frac{ab\cdot b} a =|a|+|b|[/math] [br][br]In this example, each square root written must have a calculable value. This means that their internals are worth at least 0. Therefore, we can perform multiplications of square roots with certainty. So far, we do not know anything about the values of [i]parameters a[/i] and [i]b[/i], so we need absolute value signs.