Indices examples

Example 1

What is $2^0$ in a simpler form?

Answer:

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?

Answer:

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

Answer:

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 $1^9$

Answer:

Try simpler numbers you know

We know

$2^2=2times2=4$

Therefore

$1^9=1times1times1times1times1times1times1times1times1$

$=1$

Therefore

$1^9=1$

Example 5

What is $(1\2/5)^3$

Answer:

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

$6x^3y$