Question

A cube with side length 4p is stacked on another cube with side length 2q^2. What is the total volume of the cubes in factored form?

Answer 1

Volume of the cube with side 4p = 4p x 4p x 4p = 64p³

Volume of the cube with side 2q² =  2q² x  2q² x  2q² = 8q⁶

Total Volume = 64p³ +  8q⁶

Total Volume = (4p)³ + (2q²)³

Total Volume = (4p + 2q²)( ( 4p)² - (4p)(2q²) + (2q²)²)

Total Volume = (4p + 2q²)( 16p² - 8pq² + 4q⁴)

 
Answer:  (4p + 2q²)( 16p² - 8pq² + 4q⁴)

Answer 2

we know that

The volume of a cube is equal to

$V=b^{3}$

where

b is the length side of the cube

Step 1

Find the volume of the cube $1$

we have

$b=4p\ units$

substitute in the formula

$V1=(4p)^{3}=64p^{3}\ units^{3}$

Step 2

Find the volume of the cube $2$

we have

$b=2q^{2}\ units$

substitute in the formula

$V2=(2q^{2})^{3}=8q^{6}\ units^{3}$

Step 3

Find the total volume

Adds volume 1 and volume 2

$V=64p^{3}\ units^{3}+8q^{6}\ units^{3}$

$V=(64p^{3}+8q^{6})\ units^{3}$

Step 4

Find the volume in factored form

we know that

The sum of cubes is equal to

$a^{3} +b^{3} =(a+b)(a^{2} -ab+b^{2})$

$a=4p\ units$

$b=2q^{2}\ units$

substitute

$(64p^{3}+8q^{6})=(4p+2q^{2})((4p)^{2}-(4p)(2q^{2})+(2q^{2})^{2}$

$=(4p+2q^{2})(16p^{2}-8pq^{2}+4q^{4})$

therefore

the answer is

$(4p+2q^{2})(16p^{2}-8pq^{2}+4q^{4})\ units^{3}$