Further roots

Further roots to explain $x^(a/b)$

Always split the power into a root and a power.

$x^(a/b)=x^((a\ \m\u\l\t\i\p\l\i\e\d\ \by\ \1/b))=(x^a)^(1/b)=rootb(x^a)$

Or

$x^(a/b)=x^((1/b\ \m\u\l\t\i\p\l\i\e\d\ \by\ \a))=(x^(1/b))^a=(rootbx)^a$

Just remember:

A plants power is form above the roots are at the bottom, Memory text

Example 1

What is $10^(2/10)$ ?

$10^(2/10) =(\root10\10)^2$ and reads as

Work out what we multiply by itself $10$ times to get $10$ , then multiply the answer by itself $2$ times.

In this example

The number we multiply by itself $10$ times to get $10=1.259$ (to find this see logarithms)

Then multiply $1.259$ by itself $2$ times.

$1.259xx1.259=1.585$

So $10^(2/10)=1.585$

Example 2

Break any fraction up and use simple numbers

$8^(2/3)=8^((2)times(1/3)$

This can be rewritten as

$8^((2)\times(1/3))=(8^2)^(1/3)=(64)^(1/3)=root3\64=4$

This is worded as multiply $8$ by itself $2$ times, then workout what we multiply by itself $3$ times to get this answer. The answer is $4$ .

Or (working either way)

$8^((1/3)times(2))=(8^(1/3))^2=(root3(8))^2=(2)^2=4$

This is worded as: work out what we multiply by itself $3$ times to get $8$ , then multiply the answer by itself $2$ times.

So $8^(2/3)=(root3\8)^2$

Example 3

Simplify $root4\(x^4)$

As follows:

$root4\(x^4)$ is the same as $x^(4/4)$

Therefore $x^1$

Answer: The simplification of $root4\(x^4)=x$

Example 4

What is $27^(4/3)$ ?

$27^(4/3)=27^(4times(1/3))=root3((27^4))=root3((531441))=81$

Or we could do

$27^(4/3)=27^((1/3)times4)=(root3\27)^4=(3)^4=81$

The answer is the same but in this case the second calculation was much easier.