# Square and Cube Roots Using Factor trees

Lesson

In Trees with Multiple Branches we discussed how prime factors can be used to find highest common factors and lowest common multiples. We can also use prime factors to find square and cube roots.

Let's look at factor tree and discuss how.

## Square roots

##### question 1

Evaluate: $\sqrt{1444}$1444

Think: Let's look for factors of $1444$1444 (it's most helpful if we can find numbers that make perfect squares).

So $1444$1444 can be written as $2\times2\times19\times19$2×2×19×19. In index notation, we would write this as $2^2\times19^2$22×192

Do:

 $\sqrt{1444}$√1444 $=$= $2\times19$2×19 $=$= $38$38

## Cube Roots

##### question 2

Evaluate: $\sqrt[3]{1728}$31728

Think

In index notation, we would write this as $3^3\times2^6$33×26

Since we need to find the cube root, we need to divide our groups of factors by $3$3.

Do:

 $\sqrt[3]{1728}$3√1728 $=$= $3\times2^2$3×22 $=$= $12$12

#### Worked Examples

##### Question 1

Use a factor tree to help find the $\sqrt{196}$196

a) Write $196$196 as a product of prime factors in index notation.

b) Hence find the $\sqrt{196}$196

##### Question 2

Given $85184=11^3\times2^6$85184=113×26

Find $\sqrt[3]{85184}$385184

##### Question 3

We want to find $\sqrt[3]{729}$3729

a) Write $729$729 as the product of three identical

b) Hence evaluate $\sqrt[3]{729}$3729