### Video Transcript

𝑥 multiplied by 10 to the power of
five plus 𝑥 multiplied by 10 squared is equal to 123123, where 𝑥 is a number. Calculate 𝑥 multiplied by 10 to
the power of five minus 𝑥 multiplied by 10 to the power of two, giving your answer
in standard form.

This does look really tricky. But the important thing to spot is
that 𝑥 is the same in both parts of this sum. Since 𝑥 is the same, we know it
must have the digits one, two, and three. But how do we decide whether 𝑥 is
123, 12.3, 1.23, or even 0.123? Well, these numbers are in standard
form. When numbers are in standard form,
we write them as some number multiplied by 10 to the power of some other number.

The power represented by the letter
𝑏 in this example is an integer, is a whole number. And 𝑎 has to be greater than or
equal to one and less than 10. We should assume then that 𝑥 must
be equal to 1.23 since it satisfies these conditions. Let’s check what that means for the
first number: 𝑥 multiplied by 10 to the power of five is 1.23 multiplied by 10 to
the power of five.

10 to the power of five is 10
multiplied by 10 multiplied by 10 multiplied by 10 multiplied by 10 which is
100000. So 1.23 multiplied by 10 to the
power of five is 1.23 multiplied by 100000 which is 123000. That also means that 𝑥 multiplied
by 10 squared is 1.23 multiplied by 10 squared. 10 squared is 10 multiplied by
itself. So it’s 100. And that means that 1.23 multiplied
by 100 is 123. If we add these two values 123000
and 123, we actually get the answer we need: 123123.

To answer this question then, we
are going to need to subtract the two values. We’re going to subtract 123 from
123000. Let’s consider the column
method. We can’t do zero take away
three. So we borrow from the first digit
to the left which is not zero. Here that’s three. When we borrow from the three, it
becomes two and the digit immediately to its right becomes 10.

We then borrow from the 10 and it
becomes nine and the digit immediately to its right becomes 10. Then again, we borrow from that 10
and it becomes nine. And the digit that we’re interested
in — the one on the furthest to the right — is 10. 10 minus three is seven. Nine minus two is seven. Nine minus one is eight. And then, two minus zero is
two. Two minus zero is two. And one minus zero is one.

And this means that 𝑥 multiplied
by 10 to the power of five minus 𝑥 multiplied by 10 squared is 122877. But we’re not quite finished. We need to give our answer in
standard form. Remember we said that when we put a
number in standard form, whatever we multiply by 10 to some power must be between
one and 10. So our number is going to become
1.22877 multiplied by 10 to some integer.

To work out what this integer is,
we need to figure out how many times we need to multiply 1.22877 by 10 to get our
original number. In fact, we need to move the digits
to the left five spaces. That’s the same as multiplying by
10 to the power of five. That means that our answer is
1.22877 multiplied by 10 to the power of five.