Question

the area of a rectangle, ( a = l cdot w ) is represented by the expression ( 24x^{6}y^{15} ). which could be the dimensions of the rectangle?
( 2x^{5}y^{8} ) and ( 12xy^{7} )
( 6x^{2}y^{3} ) and ( 4x^{3}y^{5} )
( 10x^{6}y^{15} ) and ( 14x^{6}y^{15} )
( 8x^{4}y^{11} ) and ( 12x^{2}y^{4} )

Explanation:

Step1: Recall the rule for multiplying monomials

When multiplying two monomials \(a x^m y^n\) and \(b x^p y^q\), we use the rules of exponents: \(x^m\times x^p=x^{m + p}\) and \(y^n\times y^q=y^{n+q}\), and the coefficients are multiplied as \(a\times b\). So the product is \(abx^{m + p}y^{n+q}\). We will apply this rule to each option.

Step2: Analyze Option 1: \(2x^{5}y^{8}\) and \(12xy^{7}\)

• Coefficient: \(2\times12 = 24\)
• For \(x\): \(x^{5}\times x=x^{5 + 1}=x^{6}\)
• For \(y\): \(y^{8}\times y^{7}=y^{8 + 7}=y^{15}\)
• The product is \(24x^{6}y^{15}\), which matches the given area. But let's check other options to be sure.

Step3: Analyze Option 2: \(6x^{2}y^{3}\) and \(4x^{3}y^{5}\)

• Coefficient: \(6\times4=24\)
• For \(x\): \(x^{2}\times x^{3}=x^{2 + 3}=x^{5}\)
• For \(y\): \(y^{3}\times y^{5}=y^{3+5}=y^{8}\)
• The product is \(24x^{5}y^{8}

eq24x^{6}y^{15}\)

Step4: Analyze Option 3: \(10x^{6}y^{15}\) and \(14x^{6}y^{15}\)

• Coefficient: \(10\times14 = 140

eq24\)

• So this option is incorrect.

Step5: Analyze Option 4: \(9x^{4}y^{11}\) and \(12x^{2}y^{4}\)

• Coefficient: \(9\times12=108

eq24\)

• So this option is incorrect.

Answer:

A. \(2x^{5}y^{8}\) and \(12xy^{7}\)