Video Transcript
What is the length of a rectangle
whose area is six 𝑥 raised to the fifth power 𝑦 plus 27𝑥 raised to the fourth
power 𝑦 square centimeters and whose width is 𝑥𝑦 centimeters?
In this question, we are asked to
determine the length of a rectangle from an expression for its area in square
centimeters in terms of unknowns 𝑥 and 𝑦 and an expression for its width in the
same unknowns in centimeters. To answer this question, we can
start by sketching the given information onto a rectangle. We can call the length of the
rectangle 𝐿 centimeters, and we can add the expressions for the area and width to
obtain the following sketch.
We can then recall that the area of
a rectangle is equal to its length multiplied by its width. We can substitute the given
expressions for the area and width of the rectangle to form an equation. We have six 𝑥 raised to the fifth
power times 𝑦 plus 27𝑥 raised to the fourth power 𝑦 is equal to 𝐿 times
𝑥𝑦. Since we want to find an expression
for the length of the rectangle, we want to isolate 𝐿 on one side of the
equation. To do this, we need to divide the
equation through by 𝑥𝑦.
Before we do this, it is worth
noting that both 𝑥 and 𝑦 are nonzero, since if either value was zero, then the
width of the rectangle would be zero centimeters. Therefore, we can divide both sides
of the equation through by 𝑥𝑦 to obtain six 𝑥 raised to the fifth power 𝑦 plus
27𝑥 raised to the fourth power 𝑦 all over 𝑥𝑦 is equal to 𝐿.
We can now note that on the
left-hand side of the equation, we are dividing a polynomial by a monomial. To simplify this division, we want
to divide every term in the numerator by the denominator separately. Splitting the division over each
term gives us the following equation. It is worth noting that we can
simplify the left-hand side of the equation by using the quotient rule for
exponents. However, we can start by canceling
the shared factor of 𝑦 in the numerator and denominator of each term, since we know
that 𝑦 is nonzero.
We can follow a similar process to
simplify the division by 𝑥. Or we can apply the quotient rule
for exponents, which tells us 𝑥 raised to the power of 𝑚 over 𝑥 raised to the
power of 𝑛 is equal to 𝑥 raised to the power of 𝑚 minus 𝑛, provided that 𝑥 is
nonzero. We can rewrite the 𝑥 in the
denominator as 𝑥 raised to the first power so that we can apply this rule to each
term. We obtain six 𝑥 raised to the
power of five minus one plus 27𝑥 raised to the power of four minus one is equal to
𝐿. Finally, we can evaluate the
expressions in the exponents and add the units of centimeters onto our length to
conclude that the length of the rectangle is six 𝑥 raised to the fourth power plus
27𝑥 cubed centimeters.
