Video Transcript
Given that three 𝑥 to the power of
three over two minus four equals one, find the value of 𝑥. If necessary, give your answer to
three decimal places.
So, the first thing we want to do
with this equation is have our 𝑥 term on its own on the left-hand side. So what we’re going to do is we’re
gonna add four to each side of the equation. And when we add four to each side
of the equation, what we’re gonna be left with is three 𝑥 to the power of three
over two equals five. And then, what we’re gonna do is
divide each side of the equation by three. So, we get 𝑥 to the power of three
over two is equal to five over three or five-thirds.
So, now, in order to solve the
problem and find the value of 𝑥, what we need to do is have a look at the
exponent. So, we’ve got 𝑥 to the power of
three over two. Well, what does this mean? Well, we can have a quick look at
an exponent rule. Well, if we have 𝑥 to the power of
𝑎 over 𝑏, this is gonna be equal to the 𝑏th root of 𝑥, to the power of 𝑎, or
the 𝑏th root of 𝑥 to the power of 𝑎.
So, we can actually see that you
could write it either way cause we can either deal with the root first or the
exponent first. So, what we’re gonna do first is
we’re gonna rewrite 𝑥 to the power of three over two. And we can rewrite this as the
square root of 𝑥 cubed. And we’ve chosen to do it this way
just because this is probably easier to start off with. So, now, to use inverse operations,
the first thing we’re gonna do is square each side of the equation. And it’s worth noting that if we
square a fraction, so, for instance, if we have 𝑎 over 𝑏 all squared, then this is
the same as 𝑎 squared over 𝑏 squared. So, you square the numerator and
square the denominator.
Well, in that case, if we square
five, we get 25. And if we square three, we get
nine. So, we get now 𝑥 cubed is equal to
25 over nine. So then, what we need to do is take
the cube root of both sides of the equation cause, again, we’re gonna use the
inverse operation. So, this’s gonna leave us with 𝑥
is equal to the cube root of 25 over nine. Well, if we type this into our
calculator, what we’re gonna get is 𝑥 is equal to 1.405721109.
But have we finished here? Well, no, we need to check what
accuracy the question wants us to leave our answer in. Well, the question says it wants
our answer left to three decimal places. Well, if we take a look at the
third decimal place, this is where we’ve got five. And then, the number to the right
of this is our deciding number. And we can see that this is a
seven. And because it’s a seven, it means
that it’s greater than five or five and greater. And if it’s five and greater, then
what we do is we round up our number to the left of this digit. So, when we do this, we get 𝑥 is
equal to 1.406. And this is, as we said, to three
decimal places.
Well, as we said when we were
looking at what 𝑥 to the power of 𝑎 over 𝑏 was, we can have it written in two
ways. To solve this problem, we used the
second way. However, just to double check, we
could use the first way of rewriting this. And that would be that the square
root of 𝑥 all cubed is equal to five over three. So, using this method, what we
would do first is take the cube root of both sides of the equation. So, that leaves us with the square
root of 𝑥 is equal to the cube root of five over three.
So then, what we do is we’d square
both sides of the equation to get our final answer. So, what we get is 𝑥 is equal to,
and then we’ve got the cube root of five over three all squared. Now, we can pop this into our
calculator. And this would give us our
answer. Or using the rule that we looked at
earlier with our exponents, we could see that it’ll be 𝑥 is equal to five over
three to the power of two over three. And we could type this into our
calculator. And either way, we get the correct
answer, which is 𝑥 is equal to 1.406. And that’s to three decimal
places.
