8th Grade - Square Roots and Cube Roots

Introduction

In the Laws of Exponents lesson, we learned how to raise a number to a power. Finding a square root of that power is a reversal of that operation.
$\sqrt{}$ is called a radical sign.
Any expression with radical sign values is called a Radical Expression.
In $\sqrt[3]{27}$ , 27 is called a radicand, and 3 is called a root number, also sometimes referred to as an index.

$3^{2}$ is 9, the square root of 9 represented as $\sqrt{9}$ is 3.
If a radical number has no root number or index, it can be considered a square root.
The square root of an x is nothing but $x^{\frac{1}{2}}$

3 raised to the power of 3, also written as $3^{3}$ is 27. The reversal of that is the cube root of 27, also written as $\sqrt[3]{27}$ is 3.

Exponents are nothing but repeated multiplication. Roots are the exact opposite of that.
$3 \times 3 = 9$ , and $- 3 \times - 3$ is also 9. So it is safe to say $\sqrt{9}$ is $\pm 3$ (as we are not sure if 9 results from the multiplication of two positive 3s or two negative 3s.
A number multiplied by itself can never have a negative result. If a number is positive, its square is always positive. If a number is negative, its square will also be a positive number. So it is safe to say $\sqrt{- 9}$ is not a real number.

To find the root of the non-perfect square, break the radicand and see if any of the factors are a perfect square.
For example, $\sqrt{50}$ can be written as $\sqrt{25 \times 2}$ , which can be further simplified as $5 \sqrt{2}$ .
The square root of a prime number is an irrational number.
To approximate the square root of a non-perfect square number, we can use the shortcut formula $\sqrt{x} \approx \frac{x + y}{2 \sqrt{y}}$ where $y$ is the nearest perfect square number to $y$ .
To approximate the cube root of a non-perfect cube number, we can use the shortcut formula $\sqrt[3]{b} \approx \frac{2 b + a}{2 a + b} \sqrt[3]{a}$ where $a$ is the nearest perfect cube number to $b$ .

$\sqrt{36}$ is $\pm 6$ as $6 \times 6$ , and $- 6 \times - 6$ both equals to 36.
$\sqrt[3]{64}$ is 4 as 4 multiplied by itself 3 times also written as $4^{3}$ is 64.
$\sqrt[5]{32}$ is 2 as $2^{5}$ is 32.

A positive number will always have two square roots, one positive and one negative.
$\sqrt{1}$ is $\pm 1$
The square root of a prime number is an irrational number.
$\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} . \sqrt{2} = 6 . \sqrt{2} = 6 \sqrt{2}$
$\sqrt[3]{4} = 4^{\frac{1}{3}}$
$\sqrt{m^{2}} = \pm m$
$\sqrt{x^{3}} = \sqrt{x^{2} . x} = x \sqrt{x}$

In case you want a root of the non-perfect square, break the radicand and see if any of the factors are a perfect square.