# Indices examples

Example 1

What is 2^0 in a simpler form?

We have two ways to remember this

## Method 1

Remember the picture "it doesnt matter how many people there are there is only one sun" which means

2^0 is therefore =1

or

## Method 2

Try simple numbers you know first

(10^2)/(10^2)=100/100=1

and also

(10^2)/(10^2)=10^2times10^-2=10^0 must=1

therefore 2^0=1

Example 2

What is 49^(-1/2) in a simpler form?

Try simpler numbers we know first

We know

10^-2=1/(10^2)

so

49^(-1/2)=1/(49^(1/2))

and we also know the 1/2 denotes that it is root 2

So

49^(1/2) is sqrt49

and

sqrt49=7

Therfore

1/49^(1/2)=1/sqrt49=1/7

So the answer =1/7

Example 3

What is 64^(1/3) in a simpler form

## Method 1

64^(1/3) actually reads

Work out what we multiply by itself 3 times to get 64

Which would be xtimesxtimesx=64

this is 4times4times4=64

Therefore 64^(1/3)=4

## Method 2

Try simpler numbers you know

You might recognise that

9^(1/2)=sqrt9=3

and

27^(1/3)=root(3)(27)=3

Therefore

64^(1/3)=root(3)(64)=4

Example 4

What is 1^9

Try simpler numbers you know

We know

2^2=2times2=4

Therefore

1^9=1times1times1times1times1times1times1times1times1

=1

Therefore

1^9=1

Example 5

What is (1\2/5)^3

The first thing to do is turn

1\2/5 into a fraction

See mammoth memory fractions

1\2/5=7/5

So (1\2/5)^3 becomes (7/5)^3

"Try simple numbers you know first"

Try (4/2)^3=2^3=2times2times2=8

is this the same as

4^3/2^3=(4times4times4)/(2times2times2)

=(cancel4\ ^2timescancel4\ ^2timescancel4\ ^2)/(cancel2timescancel2timescancel2)

=2times2times2=8

Yes it is

Therefore

(7/5)^3=7^3/5^3

=(7times7times7)/(5times5times5)

=343/275

1.247 to 3 decimal places.

Example 6

What is 16^(3/2)

To tackle this we "always split the power into a root and a power"

16^(3/2)= either (16^3)^(1/2) or (16^(1/2))^3

Lets go for

(16^(1/2))^3

Which is (root(2)(16))^3

Which is 4^3

and this is 4times4times4

=16times4

=64

Example 7

Simplify (4x^-2)/y^0

We must remember the picture

"It doesn't matter how many people there are there is only one sun" which means

y^0=1

Therefore

(4x^-2)/y^0

= (4x^-2)/1

Which is just 4x^-2

But "trying a simpler number we know first"

10^-2=1/(10^2)

Therefore

4x^-2

is the same as

4/x^2

Example 8

Simplify 18x^4y^5div3xy^4

This would be the same as

(18x^4y^5)/(3xy^4)

and can be re-written as

(18timesxtimesxtimesxtimesxtimesytimesytimesytimesytimesy)/(3xtimesytimesytimesytimesy)

and if we cancel out we get

(18timesxtimesxtimesxtimescancelxtimesytimescancelytimescancelytimescancelytimescancely)/(3timescancelxtimescancelytimescancelytimescancelytimescancely)

=(18timesxtimesxtimesxtimesy)/3

=(18x^3y)/3

Or

6x^3y