# 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 $\sqrt{27}$, 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 $\sqrt{27}$ is 3.

## Determining if the Root is Positive or Negative

• Exponents are nothing but repeated multiplication. Roots are the exact opposite of that.
• , and is also 9. So it is safe to say $\sqrt{9}$ is $±3$ (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 , 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 $\sqrt{b}\approx \frac{2b+a}{2a+b}\sqrt{a}$ where $a$ is the nearest perfect cube number to $b$.

## Examples

• $\sqrt{36}$ is $±6$ as , and both equals to 36.
• $\sqrt{64}$ is 4 as 4 multiplied by itself 3 times also written as ${4}^{3}$ is 64.
• $\sqrt{32}$ 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 $±1$
• The square root of a prime number is an irrational number.

## 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.