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:
| $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√4×3√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 |