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The symbol |\sqrt{\phantom{2}}| is called a radical.
The number or algebraic expression under the radical is called the radicand.
The degree of a root, located at the top left of the radical, is called the index.
||\sqrt[\Large{\color{#3b87cd}{\textbf{index}}}]{\color{#ec0000}{\textbf{radicand}}}= \color{#3a9a38}{\textbf{root}}||
|\sqrt{16}| is the square root of |16| and it is equal to |4| because |4^2=16.|
|\sqrt[\large3]{125}| is the cube root of |125| and it is equal to |5| because |5^3=125.|
|\sqrt[\large4]{1296}| is the fourth root of |1296| and it is equal to |6| because |6^4=1296.|
Generalizing, we get:
The square root of a number |y| corresponds to a positive real number |x| which, squared, equals |y.| ||\sqrt{y} = x\ \Longleftrightarrow\ x^2=y||
In summary, taking a square root and squaring are inverse operations.
The square root of |49| is |7,| because |7| squared equals |49.| ||\sqrt{49} = 7\ \Longleftrightarrow\ 7^2=49||
A square root can be used to isolate a variable that is squared in an algebraic equation. It's very useful for finding missing measurements in plane figures from their area.
||\begin{align} c^2 &= 184.96\\ \color{#ec0000}{\sqrt{\color{black}{c^2}}}&=\color{#ec0000}{\sqrt{\color{black}{184.96}}}\\ c &= 13.6\end{align}||
In the set of real numbers |(\mathbb{R}),| we can't calculate the square root of negative numbers.
This is because of the sign rule that states that any number squared equals a positive number, regardless if it is positive or negative itself. For example, |5^2= 5 \times 5 = 25| and |(-5)^2=-5 \times -5 = 25| too. Therefore, in |\mathbb{R},| it is impossible to calculate the square root of |-25,| since there is no number that, when multiplied by itself, equals |-25.|
According to the definition, the square root of a number is always positive. However, when solving second degree equations, it's important to consider both solutions, not just the positive one.
In other words, even if the square root of |81| is |9| (positive), the equation |x^2=81| has 2 solutions: |9| and |-9.|
What are the solutions of the following equation? ||8x^2=450||
First divide by |8| to isolate |x^2.| ||\begin{align} 8x^2&=450\\ \color{#ec0000}{\dfrac{\color{black}{8x^2}}{\boldsymbol{8}}} &= \color{#ec0000}{\dfrac{\color{black}{450}}{\boldsymbol{8}}}\\ x^2&=56.25\end{align}||We are now looking for a number |x| which, when squared, equals |56.25.| We must therefore calculate the square root of |56.25.| ||\sqrt{56.25}=7.5|||7.5| is a solution of the equation. However, it's not the only solution, since |-7.5| is also a valid solution. The following validates both of these solutions.||\begin{gather}8\boldsymbol{\color{#3a9a38}{x}}^2=450\\\swarrow\searrow\\\begin{aligned}8(\boldsymbol{\color{#3a9a38}{7.5}})^2&\overset{?}{=}450 &8(\boldsymbol{\color{#3a9a38}{-7.5}})^2&\overset{?}{=}450\\[3pt]8(56.25)&\overset{?}{=}450&8(56.25)&\overset{?}{=}450\\[3pt]450&=450&450&=450\end{aligned}\end{gather}||Answer: The solutions of the equation are |7.5| and |-7.5.|
Whenever we solve a quadratic equation, to avoid forgetting the 2nd solution, we use the symbol |\pm| when we take the square root.||\begin{aligned} x^2&=56.25\\ x&=\color{#ec0000}{\boldsymbol\pm\sqrt{\color{black}{56.25}}}\\ &\swarrow\ \searrow\end{aligned}\\ \begin{alignat}{1}\!\!\!\!\!\! x_1 &=\color{#ec0000}{\boldsymbol +\sqrt{\color{black}{56.25}}} \qquad x_2&=\color{#ec0000}{\boldsymbol -\sqrt{\color{black}{56.25}}}\\ &=\color{#ec0000}{\boldsymbol +}\,7.5 &=\color{#ec0000}{\boldsymbol -}\,7.5 \end{alignat}||
The cube root of a number |y| corresponds to a real number |x| which, cubed, equals |y.| ||\sqrt[\large{3}]{y} = x\ \Longleftrightarrow\ x^3=y||
In summary, taking a cube root and cubing a number are inverse operations.
The cube root of |125| is |5,| because |5| cubed equals |125.| ||\sqrt[\large{3}]{125} = 5\ \Longleftrightarrow\ 5^3=125||
The cube root can be used to isolate a variable that is cubed in an algebraic equation. It's very useful for finding missing measurements in solids from the volume.
||\begin{align} c^3 &= 2197\\ \color{#ec0000}{\sqrt[\large{3}]{\color{black}{c^3}}}&=\color{#ec0000}{\sqrt[\large{3}]{\color{black}{2197}}}\\ c &= 13\end{align}||
Unlike the square root, it is possible to calculate the cube root of a negative number. In fact, within the set of real numbers, it's always possible to calculate an odd-degree root, whether the radicand is positive or negative, whereas for even-degree roots, the radicand must be positive.
||\begin{align}(\boldsymbol{\color{#ec0000}{-}}4)^3 &= \boldsymbol{\color{#ec0000}{-}}4 \times \underline{\boldsymbol{\color{#ec0000}{-}}4 \times \boldsymbol{\color{#ec0000}{-}}4}\\[2pt] &=\underline{\boldsymbol{\color{#ec0000}{-}}4\times \boldsymbol{\color{#3a9a38}{+}}16}\\[2pt] &= \boldsymbol{\color{#ec0000}{-}}64\end{align}||Therefore, the cube root of |-64| is |-4| since the cube of |-4| is |-64.| ||\sqrt[\large{3}]{-64} = -4\ \Longleftrightarrow\ (-4)^3=-64||