# Question Video: Simplifying Polynomials by Division to Express the Length of a Rectangle Mathematics • 9th Grade

What is the length of a rectangle whose area is 10𝑥⁶𝑦 cm² and whose width is 5𝑥𝑦 cm?

02:52

### Video Transcript

What is the length of a rectangle whose area is 10𝑥 raised to the sixth power 𝑦 square centimeters and whose width is five 𝑥𝑦 centimeters?

In this question, we want to determine the length of a rectangle by using given expressions in terms of unknowns 𝑥 and 𝑦 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 𝐿 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 𝐿 times five 𝑥𝑦 is equal to 10𝑥 raised to the power of six times 𝑦. We can rewrite the equation to find an expression for 𝐿 in terms of 𝑥 and 𝑦. We divide the equation through by five 𝑥𝑦 to obtain that 𝐿 equals 10𝑥 raised to the sixth power 𝑦 all over five 𝑥𝑦.

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 𝑏 is nonzero, then 𝑏 raised to the power of 𝑚 over 𝑏 raised to the power of 𝑛 is equal to 𝑏 raised to the power of 𝑚 minus 𝑛. We can apply this to each variable separately.

It is worth noting that we know that 𝑥 and 𝑦 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 𝑦 is nonzero, we can cancel the shared factor of 𝑦 in the numerator and denominator.

Finally, we can rewrite the factor of 𝑥 in the denominator as 𝑥 raised to the first power and apply the quotient rule for exponents to get two 𝑥 raised to the power of six minus one. We can evaluate the expression in the exponent to obtain two 𝑥 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 𝑥 raised to the fifth power centimeters.