Video Transcript
Find the value of eight multiplied
by 10 to the 10th power add 7.1 multiplied by 10 to the ninth power, giving your
answer in standard form.
These two numbers are given in
standard form. This is also known as scientific
notation, meaning that they are in the form 𝑎 multiplied by 10 to the 𝑛th power,
where the absolute value of 𝑎 is greater than or equal to one and less than 10 and
𝑛 is an integer.
One way we could answer this
question is by writing these numbers so that they’re not in standard form. But we’d end up with two really
large numbers. There is a different way we can
approach this, but we first need to have both of these numbers written with the same
power of 10. We usually do this by rewriting the
number with the highest power of 10 to be the same as the power of 10 of the
lower-power number. The number with the higher power of
10 is eight multiplied by 10 to the 10th power, so we aim to rewrite this as
something multiplied by 10 to the ninth power so that it has the same power of 10 as
7.1 multiplied by 10 to the ninth power. Another way of writing 10 to the
10th power is 10 to the one plus nine power.
Now recall the exponent product
rule: 𝑥 to the 𝑚 power multiplied by 𝑥 to the 𝑛 power equals 𝑥 to the 𝑚 add 𝑛
power. We can use this to write 10 to the
one add nine power as 10 to the first power multiplied by 10 to the ninth power. 10 to the first power is just
10. So to end up with something
multiplied by 10 to the ninth power, we multiply eight by 10 to get 80 multiplied by
10 to the ninth power.
So now we have two numbers to add
together, both are raised to the 10 to the ninth power. So now let’s add these two numbers
together. We are calculating 80 multiplied by
10 to the ninth power add 7.1 multiplied by 10 to the ninth power. Because we were able to write both
of our numbers with the same power of 10, we can add them by doing 80 add 7.1
multiplied by 10 to the ninth power. That gives us 87.1 multiplied by 10
to the ninth power.
Although this is correct for this
calculation, we can’t leave our answer like this. That’s because we were asked to
give our answer in standard form. And this isn’t in standard form,
because standard form requires the absolute value of 𝑎, which is 87.1, to be
between one and 10, which 87.1 is not. So we need to write 87.1 multiplied
by 10 to the ninth power in standard form.
To do this, we can use a similar
method to our first step, by using the exponent product rule. As 87.1 is 8.71 multiplied by 10,
we start by writing 87.1 multiplied by 10 to the ninth power as 8.71 multiplied by
10 multiplied by 10 to the ninth power. Then 10 can be written as 10 to the
first power. And now we use the exponent product
rule. This means that we can write 10 to
the first power multiplied by 10 to the ninth power as 10 to the one add nine
power. But one add nine is 10. So this is 10 to the 10th
power.
Let’s just check that this is in
standard form. The value of 𝑎 is 8.71. So the absolute value of 𝑎 is
greater than or equal to one and less than 10, so that’s ok. Also 𝑛 is 10, which is an
integer. So this number is in standard
form. And so 8.71 multiplied by 10 to the
10th power is our final answer.
