# Simplify further expressions involving algebraic roots

Lesson

## Dealing with cube roots

Expressions like $\sqrt[3]{7}$37 and $\sqrt[3]{-\frac{3}{2}}$332 where the numbers inside the radical sign are not the cube of a rational number, are unable to be expressed as rational numbers.

We can however write down rational approximations for them. For example, $\sqrt[3]{7}\approx1.913$371.913 and $\sqrt[3]{-\frac{3}{2}}\approx-1.148$3321.148.

Expressions of the form$\sqrt[3]{a}$3a where a is a cube or a whole multiple of cube can be simplified. The trick of course is to recognise these cubes whenever they occur.

Here is a table showing the first $6$6 cubes:

Useful cubes
$1^3$13 $2^3$23 $3^3$33 $4^3$43 $5^3$53 $6^3$63
$1$1 $8$8 $27$27 $64$64 $125$125 $216$216
##### Example 1

$\sqrt[3]{8}=2$38=2. Note also that $\sqrt[3]{8}=8^{\frac{1}{3}}=\left(2^3\right)^{\frac{1}{3}}=2^1=2$38=813=(23)13=21=2

##### Example 2

$\sqrt[3]{108}=\sqrt[3]{4\times27}=\sqrt[3]{4}\times\sqrt[3]{27}=3\sqrt[3]{4}$3108=34×27=34×327=334

##### Example 3

Simplify $\sqrt[3]{2}\left(2-\sqrt[3]{32}\right)$32(2332)

Then:

 $\sqrt[3]{2}\left(2-\sqrt[3]{32}\right)$3√2(2−3√32) $=$= $2\sqrt[3]{2}-\sqrt[3]{2\times32}$23√2−3√2×32 $=$= $2\sqrt[3]{2}-\sqrt[3]{64}$23√2−3√64 $=$= $2\sqrt[3]{2}-4$23√2−4 $\approx$≈ $-1.48$−1.48
##### Example 4

$8^{\frac{5}{3}}=\left(8^{\frac{1}{3}}\right)^5=\left(\sqrt[3]{8}\right)^5=2^5=32$853=(813)5=(38)5=25=32

## Fourth and higher roots

The same general principles apply for fourth and higher roots.

##### Example 5

$48\sqrt[4]{3}+\sqrt[4]{48}$4843+448

Simplification:

 $48\sqrt[4]{3}+\sqrt[4]{48}$484√3+4√48 $=$= $48\sqrt[4]{3}+\sqrt[4]{16\times3}$484√3+4√16×3 $=$= $48\sqrt[4]{3}+\sqrt[4]{16}\times\sqrt[4]{3}$484√3+4√16×4√3 $=$= $48\sqrt[4]{3}+2\times\sqrt[4]{3}$484√3+2×4√3 $=$= $48\sqrt[4]{3}+2\sqrt[4]{3}$484√3+24√3 $=$= $50\sqrt[4]{3}$504√3 $\approx$≈ $65.8$65.8