# Standard form difficult examples

Example 1

(3.2times10^-5)times(4.8times10^7)

## Method 1

Is to actually carry out the multiplication

(3.2times10^-5)times(4.8times10^7)

is the same as

3.2/(cancel10timescancel10timescancel10timescancel10timescancel10)

times4.8times10times10timescancel10timescancel10timescancel10timescancel10timescancel10

=3.2times4.8times10times10

=1,536

Now place in standard form

remember move decimal place to the left and +1 to the power

=1.536times10^3

## Method 2

An alternative way to calculate this is to use indices. What is a similar calculation? What is:

10^2times10^-3=(10times10)/(10times10times10)=1/10=10^-1

Therefore

10^2times10^-3=10^(2-3)=10^1

So

(3.2times10^-5)times(4.8times10^7)

=3.2times4.8times10^-5times10^7

=3.2times4.8times10^(-5+7)

=3.2times4.8times10^2

=3.2times4.8times10times10

=1,536

and in standard form that equals

remember move decimal place to left and +1 to the power

1.536times10^3

Example 2

sqrt((6.8times10^9-3.8times10^8)/(6.8times10^9times3.8times10^8))

## Method 1

Is to actually carry out each calculation but achieve this step by step

6.8times10^9-3.8times10^8

is the same as

6.8times10times10times10times10times10times10times10times10times10

-3.8times10times10times10times10times10times10times10times10

=6,800,000,000-380,000,000

which is

 1 = 6, 8 0 0, 0 0 0, 0 0 0 4 - cancel3 8 0, 0 0 0, 0 0 0 6, 4 2 0, 0 0 0, 0 0 0

and

6.8times10^9times3.8times10^8

is the same as

6.8times10times10times10times10times10times10times10times10times10

times3.8times10times10times10times10times10times10times10times10

and the same as

6.8times3.8times10^17

which is

25.84times10^17

so the calculation becomes

sqrt((6,420,cancel(000),cancel(000))/(2584,000,000,000,cancel(000),cancel(000)))

or

sqrt((6,420)/(2584,000,000,000))

=sqrt0.00000000248452

=0.00004984

or in standard form

remember move decimal to right and -1 to the power

4.984times10^-5

## Method 2

An alternative way is to add or take the indices.

6.8times10^9-3.8times10^8

We have to make the powers the same when we subtract.

Remember moving decimals to the right and -1 to the power.

so

6.8times10^9=68times10^8

this becomes

68times10^8-3.8times10^8

which is

(68-3.8)times10^8

=64.2times10^8

Now for the bottom line

6.8times10^9times3.8times10^8

which is

6.8times3.8times10^9times10^8

this becomes

6.8times3.8times10^17

25.84times10^17

The whole sum becomes

sqrt((64.2timescancel(10^8))/(25.84times10^(cancel(\ 17)9))

which is

sqrt((64.2)/(25.84times10^9))

or

sqrt(64.2/25840000000)

=0.0000498.......

or

4.98.....times10^-5

Example 3

(5times10^3)/(9times10^-5)

## Method 1

Actually carry out the division of this sum

5times10^3=5times10times10times10=5000

9times10^-5=9/(10times10times10times10times10)

=0.00009

so the sum becomes

5000/0.00009

=55,555,555.dot5

In standard form this would be:

remember moving decimal place to the left and +1 to the power

5.dot5times10^7

## Method 2

The alternative way is to add or subtract the indices

(5times10^3)/(9times10^-5)

is the same as

(5times10^3times10^5)/9

which is

5/9times10^8

which is

0.5dot5times10^8

but in standard form

remember moving the decimal to the right and -1 to the power

so

0.5dot5times10^8

becomes

5.dot5times10^7