Video Transcript
Simplify the square root of three
squared multiplied by the negative square root of three to the power of 13 over the
square root of three cubed.
The first thing we notice is how
each parentheses contains the square root of three, with the exception of the second
parentheses in the numerator, which contains the negative square root of three. We could use properties of
exponents to factor the negative one out of the second parentheses, specifically
using the power of a product rule. This rule states that the product
of 𝑎 and 𝑏 to the power of 𝑛 is equal to the product of 𝑎 to the power of 𝑛
times 𝑏 to the power of 𝑛 for nonzero real numbers 𝑎 and 𝑏 and integer 𝑛.
We can write the negative square
root of three to the power of 13 as negative one times the square root of three to
the power of 13, then, following this rule, negative one to the power of 13 times
the square root of three to the power of 13. So the numerator of our expression
is now the square root of three squared times negative one to the power of 13 times
the square root of three to the power of 13. The denominator has not been
changed.
Next, we will evaluate negative one
to the power of 13. We know that negative one squared
equals positive one, negative one cubed equals negative one, negative one to the
power of four equals one, negative one to the power of five equals negative one, and
so on. As the pattern reveals, any even
power of negative one equals one, and any odd power of negative one equals negative
one. Therefore, negative one to the
power of 13 equals negative one.
Because multiplication is
commutative and associative, we can move the factor of negative one to the front of
the rational expression. Next, we will simplify the
numerator using the product rule for exponents. We recall that this rule states
that 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, where 𝑎 is a nonzero real number and 𝑚 and 𝑛 are
integers.
Using the product rule, we can
write the numerator as the square root of three to the power of two plus 13. So we now have negative one times
the quotient of the square root of three to the power of 15 over the square root of
three cubed. Now we recall the quotient rule for
exponents, which states that the quotient of 𝑎 to the power of 𝑚 over 𝑎 to the
power of 𝑛 is equal to 𝑎 to the power of 𝑚 minus 𝑛, where 𝑎 is a nonzero real
number and 𝑚 and 𝑛 are integers. This means we have the square root
of three to the power of 15 minus three. So the new power of the square root
of three is 12.
Now in order to simplify the square
root of three to the power of 12, we will use the product of squares along with the
following rule. For 𝑎 greater than zero, the
square root of 𝑎 squared is equal to 𝑎. So next, we use an expanded form of
the product rule in reverse to write out the square root of three to the power of 12
as the product of six factors of the square root of three squared. Each square root of three squared
equals three. In short, we can now write the
expression as negative one times three to the power of six. And three to the power of six is
729. Finally, we multiply negative one
and 729.
In conclusion, we have shown that
the square root of three squared multiplied by the negative square root of three to
the power of 13 over the square root of three cubed simplifies to negative 729.
