Simplify further expressions involving algebraic roots

Dealing with cube roots

Expressions like $\sqrt[3]{7}$³√7 and $\sqrt[3]{-\frac{3}{2}}$³√−32​ 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, $\sqrt[3]{7}\approx1.913$³√7≈1.913 and $\sqrt[3]{-\frac{3}{2}}\approx-1.148$³√−32​≈−1.148.

Expressions of the form$\sqrt[3]{a}$³√a 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

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

Example 2

$\sqrt[3]{108}=\sqrt[3]{4\times27}=\sqrt[3]{4}\times\sqrt[3]{27}=3\sqrt[3]{4}$³√108=³√4×27=³√4׳√27=3³√4

Example 3

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

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=(³√8)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}$48⁴√3+⁴√48

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

 

Worked Examples

Question 1

Question 2

Question 3