## 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.

## Square Roots

- ${3}^{2}$ 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 ${x}^{\frac{1}{2}}$

## Cube Roots

- 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.

## Determining if the Root is Positive or Negative

- 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.

## The Root of Non-perfect Squares

- 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 $\sqrt[3]{b}\approx \frac{2b+a}{2a+b}\sqrt[3]{a}$ where $a$ is the nearest perfect cube number to $b$.

## Examples

- $\sqrt{36}$ 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.

## Cheat Sheet

- 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}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$
- $\sqrt{{m}^{2}}=\pm m$
- $\sqrt{{x}^{3}}=\sqrt{{x}^{2}.x}=x\sqrt{x}$

## Blunder Area

- 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.

- Rishi Jethwa
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