NZ Level 8 (NZC) Level 3 (NCEA) [In development]
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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)$32(2332) $=$= $2\sqrt[3]{2}-\sqrt[3]{2\times32}$23232×32
  $=$= $2\sqrt[3]{2}-\sqrt[3]{64}$232364
  $=$= $2\sqrt[3]{2}-4$2324
  $\approx$ $-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}$4843+448 $=$= $48\sqrt[4]{3}+\sqrt[4]{16\times3}$4843+416×3
  $=$= $48\sqrt[4]{3}+\sqrt[4]{16}\times\sqrt[4]{3}$4843+416×43
  $=$= $48\sqrt[4]{3}+2\times\sqrt[4]{3}$4843+2×43
  $=$= $48\sqrt[4]{3}+2\sqrt[4]{3}$4843+243
  $=$= $50\sqrt[4]{3}$5043
  $\approx$ $65.8$65.8

 

Worked Examples

Question 1

Question 2

Question 3

Outcomes

M8-9

Manipulate complex numbers and present them graphically

91577

Apply the algebra of complex numbers in solving problems

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