Video Transcript
Express the square of negative
square root of three cubed multiplied by negative square root of two squared in its
simplest form.
To simplify this expression, we
need to recall and apply the relevant laws of exponents. First, we apply the power of a
product rule, which states that for real numbers 𝑎 and 𝑏 and integer 𝑛, the
product of 𝑎 and 𝑏 to the power of 𝑛 is equal to 𝑎 to the power of 𝑛 multiplied
by 𝑏 to the power of 𝑛. This means we can rewrite our
expression as negative square root of three cubed to the power of two multiplied by
negative square root of two squared to the power of two.
Next, we apply the power of a power
rule, which says for real number 𝑎 and integers 𝑚 and 𝑛, 𝑎 to the power of 𝑚 to
the power of 𝑛 is equal to 𝑎 to the power of 𝑚 times 𝑛. This means we can write our
expression as negative square root of three to the power of six multiplied by
negative square root of two to the power of four.
At this point, we may be tempted to
use the product rule, which says that for real number 𝑎 and integers 𝑚 and 𝑛, 𝑎
to the power of 𝑚 multiplied by 𝑎 to the power of 𝑛 is equal to 𝑎 to the power
of 𝑚 plus 𝑛. But this would be a mistake,
because the product rule only applies when the real number bases are the same.
We note that a negative square root
is equal to negative one times the root. So we can use the power of a
product rule to rewrite our expression as negative one to the power of six
multiplied by square root of three to the power of six multiplied by negative one to
the power of four multiplied by the square root of two to the power of four.
We recall that for any even integer
𝑚, negative one to the power of 𝑚 is equal to positive one. So negative one to the power of six
equals one, and negative one to the power of four equals one. So we are left with the square root
of three to the power of six multiplied by the square root of two to the power of
four.
We recall that for an expression
with a base 𝑎, where 𝑎 is positive, the square root of 𝑎 squared is equal to
𝑎. So we can write the square root of
three to the power of six and the square root of two to the power of four as a
product of powers of two using the product rule in reverse. This gives us the square root of
three squared multiplied by the square root of three squared multiplied by the
square root of three squared multiplied by the square root of two squared multiplied
by the square root of two squared.
Since the square root of 𝑎 squared
equals 𝑎, we have three times three times three times two times two. Therefore, the square of negative
square root of three cubed multiplied by the negative square root of two squared
simplifies to 108.
