Video Transcript
What is the length of a rectangle
whose area is 10π₯ raised to the sixth power π¦ square centimeters and whose width
is five π₯π¦ centimeters?
In this question, we want to
determine the length of a rectangle by using given expressions in terms of unknowns
π₯ and π¦ for the area and width of the rectangle. We can begin by sketching the
information given. If we say that the length of the
rectangle is πΏ centimeters, then we can add this along with the width and area of
the rectangle to obtain the following figure.
We know that the area of a
rectangle is equal to the product of its length and width. So we must have that πΏ times five
π₯π¦ is equal to 10π₯ raised to the power of six times π¦. We can rewrite the equation to find
an expression for πΏ in terms of π₯ and π¦. We divide the equation through by
five π₯π¦ to obtain that πΏ equals 10π₯ raised to the sixth power π¦ all over five
π₯π¦.
We now have the quotient of two
monomials. So we can simplify this expression
by recalling that the quotient rule for exponents tells us if π is nonzero, then π
raised to the power of π over π raised to the power of π is equal to π raised to
the power of π minus π. We can apply this to each variable
separately.
It is worth noting that we know
that π₯ and π¦ are nonzero, since if they were zero, then the width of the rectangle
would be zero. We can cancel the shared factor of
five in the numerator and denominator to obtain two. Next, since π¦ is nonzero, we can
cancel the shared factor of π¦ in the numerator and denominator.
Finally, we can rewrite the factor
of π₯ in the denominator as π₯ raised to the first power and apply the quotient rule
for exponents to get two π₯ raised to the power of six minus one. We can evaluate the expression in
the exponent to obtain two π₯ raised to the fifth power.
Finally, we can recall that the
lengths are all measured in centimeters. So we can conclude that the length
of the rectangle is two π₯ raised to the fifth power centimeters.
