### Video Transcript

Without using a calculator, find an
approximate solution to 𝑥 minus two all raised to the power of three equals six,
accurate to one decimal place.

We want to take cube roots on both
sides of the equation, giving us 𝑥 minus two equals the cube root of six. Since we are not using calculators,
we need to approximate the value of the cube root of six. Since six lies between one cubed
and two cubed, we know that the cube root of six lies between one and two. In fact, since six is closer to
eight than it is to one, we know that the cube root of six lies in the right-hand
side of the interval. That is, it is greater than
1.5.

Now, we actually have to do some
calculating. We are going to try to pin down the
location of the radical by trial and improvement. Let’s start at the top end of the
interval and cube 1.9. I like to cube decimals like this
by splitting them up into quote, unquote binomials and expanding. 1.9 cubed equals one plus 0.9 all
cubed, which equals one plus three times 0.9 plus three times 0.9 squared plus 0.9
cubed. This equals one plus 2.7 plus three
times 0.81, which is 2.43, plus 0.729. This is equal to 6.859, which is a
little too big.

So let’s try 1.8 cubed. This is equal to one plus 2.4 plus
three times 0.64, which is 1.92, plus 0.512, which equals 5.832. Six lies between 1.8 cubed and 1.9
cubed. This means that the cube root of
six lies between 1.8 and 1.9. We can see that six is closer to
5.832 than it is to 6.859. This leads us to suspect that the
cube root of six, rounded to one decimal place, is 1.8.

Unfortunately, we do actually have
to check. We’ll need to cube the midpoint
1.85 to see if it is greater than or less than six. This is a reasonably unpleasant
calculation, but there’s no way to avoid it. Let’s go. First, we do 185 times 185, which
is 34225. Next, we multiply this, i.e., 185
squared, by 185 again. This is 6331625. So 1.85 cubed is 6.331625, which is
bigger than six. Therefore, the cube root of six is
less than 1.85. This gives the approximation of the
cube root of six to one decimal place as 1.8. Adding two to both sides of the
equation, we get the approximate solution of 𝑥 equals 3.8, accurate to one decimal
place.