Video Transcript
Fill in the blank. The expression root three
multiplied by root three squared multiplied by root three cubed simplifies to
blank.
We need to simplify this expression
which consists of three powers of the same base, root three, multiplied
together. We can therefore apply the product
rule for exponents, which states that for real values of 𝑎, 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. Now there’s no exponent written for
the first term in the product, but we know that any number is equal to that number
to the first power. So root three is root three to the
power of one. So applying the product rule and
summing all three exponents, we have that this expression simplifies to root three
to the power of one plus two plus three, which is root three to the sixth power. We now know that we have six lots
of root three multiplied together, which we can write out longhand.
Next, we see that we can group the
terms in this product into pairs. And hence, we can express the
product as root three squared multiplied by root three squared multiplied by root
three squared. We can then recall the result that
for nonnegative 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. The expression is therefore equal
to three multiplied by three multiplied by three or three cubed, which is equal to
27. So filling in the blank, the given
expression simplifies to 27.
