Video Transcript
If 𝑥 equals root two over
three, 𝑦 equals one over root two, and 𝑧 equals root three over three, find
𝑥𝑦 squared 𝑧 to the negative two power in its simplest form.
Well, to find the value of 𝑥𝑦
squared multiplied by 𝑧 to the negative two power, we need to begin by
substituting the values for 𝑥, 𝑦, and 𝑧 into the expression. Doing so gives us root two over
three multiplied by one over root two squared multiplied by root three over
three to the negative two power.
Let’s simplify what we have
inside this parentheses by multiplying root two over three by one over root
two. That gives us root two over
three root two. Let’s now calculate root two
over three root two squared. That’s going to be root two
over three root two multiplied by root two over three root two. Then, multiplying the
numerators together, we get two. And multiplying the
denominators together, we get nine multiplied by two. That’s because we can firstly
multiply the threes to give nine and then the root twos to give us two. Because we know that root two
multiplied by root two just gives us two.
Now, nine multiplied by two
gives us 18. And then we can divide both the
numerator and denominator by two to simplify this. That gives us one over
nine. Another approach we could use
to evaluate this would be to cancel the root two on the numerator with the root
two on the denominator and evaluate one-third squared.
Now, let’s consider root three
over three to the negative two power. We firstly recall the law for
negative exponents, which states that 𝑎 to the negative 𝑛th power is one over
𝑎 to the 𝑛th power, for 𝑎 in the set of real numbers without zero and 𝑛 an
integer. This means that we can write
root three over three to the negative two power as one over root three over
three to the second power.
Let’s evaluate the denominator,
root three over three to the second power. This is root three over three
multiplied by root three over three. Then, multiplying the
numerators, root three by root three, gives us three. And multiplying the
denominators, three multiplied by three, gives us nine. And three over nine simplifies
to one over three. So now we’ve calculated 𝑥𝑦
squared 𝑧 to the negative two power to be one over nine multiplied by one over
one over three.
Now, we can actually rewrite
one over one-third, because this is simply one divided by a third, and that’s
just three. So we have one over nine
multiplied by three. That’s three over nine, which
is one-third.
