Standard form difficult examples
Example 1
Work out the following giving your answer in standard form
(3.2×10-5)×(4.8×107)
Method 1
Is to actually carry out the multiplication
(3.2×10-5)×(4.8×107)
is the same as
3.210×10×10×10×10
×4.8×10×10×10×10×10×10×10
=3.2×4.8×10×10
=1,536
Now place in standard form
remember move decimal place to the left and +1 to the power
=1.536×103
Method 2
An alternative way to calculate this is to use indices. What is a similar calculation? What is:
102×10-3=10×1010×10×10=110=10-1
Therefore
102×10-3=102-3=101
So
(3.2×10-5)×(4.8×107)
=3.2×4.8×10-5×107
=3.2×4.8×10-5+7
=3.2×4.8×102
=3.2×4.8×10×10
=1,536
and in standard form that equals
remember move decimal place to left and +1 to the power
1.536×103
Example 2
Work out the following giving your answer in standard format:
√6.8×109-3.8×1086.8×109×3.8×108
Method 1
Is to actually carry out each calculation but achieve this step by step
6.8×109-3.8×108
is the same as
6.8×10×10×10×10×10×10×10×10×10
-3.8×10×10×10×10×10×10×10×10
=6,800,000,000-380,000,000
which is
| 1 | |||||||||||
| = | 6, | 8 | 0 | 0, | 0 | 0 | 0, | 0 | 0 | 0 | |
| 4 | |||||||||||
| - | 3 | 8 | 0, | 0 | 0 | 0, | 0 | 0 | 0 | ||
| 6, | 4 | 2 | 0, | 0 | 0 | 0, | 0 | 0 | 0 |
and
6.8×109×3.8×108
is the same as
6.8×10×10×10×10×10×10×10×10×10
×3.8×10×10×10×10×10×10×10×10
and the same as
6.8×3.8×1017
which is
25.84×1017
so the calculation becomes
√6,420,000,0002584,000,000,000,000,000
or
√6,4202584,000,000,000
=√0.00000000248452
=0.00004984
or in standard form
remember move decimal to right and -1 to the power
4.984×10-5
Method 2
An alternative way is to add or take the indices.
Start with the top line
6.8×109-3.8×108
We have to make the powers the same when we subtract.
Remember moving decimals to the right and -1 to the power.
so
6.8×109=68×108
this becomes
68×108-3.8×108
which is
(68-3.8)×108
=64.2×108
Now for the bottom line
6.8×109×3.8×108
which is
6.8×3.8×109×108
this becomes
6.8×3.8×1017
25.84×1017
The whole sum becomes
√64.2×10825.84×10 179
which is
√64.225.84×109
or
√64.225840000000
=0.0000498.......
or
4.98.....×10-5
Example 3
Work out the following giving your answer in standard form
5×1039×10-5
Method 1
Actually carry out the division of this sum
5×103=5×10×10×10=5000
9×10-5=910×10×10×10×10
=0.00009
so the sum becomes
50000.00009
=55,555,555..5
In standard form this would be:
remember moving decimal place to the left and +1 to the power
5..5×107
Method 2
The alternative way is to add or subtract the indices
5×1039×10-5
is the same as
5×103×1059
which is
59×108
which is
0.5.5×108
but in standard form
remember moving the decimal to the right and -1 to the power
so
0.5.5×108
becomes
5..5×107
