 # 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: 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.