Video Transcript
What is the value of negative root
three to the sixth power divided by root three to the third power?
The bases in this expression are
nearly the same, but one is the negative of the other. We can’t yet apply laws of
exponents to simplify this division as the bases aren’t exactly the same. So let’s instead think about how we
can alter this expression so that the bases are identical.
In the first term, we have the
power of a product because negative root three is equal to negative one multiplied
by root three. We can therefore recall the power
of a product rule for exponents, which states that a product 𝑎𝑏 to the 𝑛th power
is equal to 𝑎 to the 𝑛th power multiplied by 𝑏 to the 𝑛th power. So we can rewrite the first term as
negative one to the sixth power multiplied by root three to the sixth power. We can then recall that negative
one to any even power is positive one. So in fact, the first term
simplifies to root three to the sixth power.
We now have exactly the same base
for both parts of the quotient. So we can simplify using the
quotient rule for exponents, which tells us that to divide powers of the same base,
we subtract the exponents. So root three to the sixth power
divided by root three to the third power is root three to the power of six minus
three, which is root three cubed. We can express this longhand as
root three multiplied by root three multiplied by root three. And then we can group the first two
terms in the product together to give root three squared multiplied by root
three.
We then recall that for nonnegative
real values of 𝑎, the square root of 𝑎 squared is equal to 𝑎. And so the square root of three
squared is equal to three. And the expression simplifies to
three root three. We found that the value of negative
root three to the sixth power divided by root three to the third power is three root
three.
