Question

Which expression is a sum of cubes?

Answer 1

Since $\sqrt[3]{32}\not\in\mathbb Z$, you can omit B and C as possible solutions.

Now,

$-64x^6y^{12}+125x^{16}y^3=(-4x^2y^4)^3+(5x^{16/3}y)^3$

but since 16 is not divisible by 3, A is not correct either.

So D is the answer. To verify:

$64x^6y^{12}+125x^9y^3=(4x^2y^4)^3+(5x^3y)^3$

Answer 2

The option D is expressed as the sum of two cubic expressions.  

Further explanation:  

Given:  

The given expressions are as follows:  

A. $-64{x^6}{y^{12}}+125{x^{16}}{y^3}$

B. $-32{x^6}{y^{12}}+125{x^{16}}{y^3}$

C. $32{x^6}{y^{12}}+125{x^9}{y^3}$  

D. $64{x^6}{y^{12}}+125{x^9}{y^3}$

Calculation:  

Step 1:  

For option A the expression $-64{x^6}{y^{12}}+125{x^{16}}{y^3}$ is simplified as follows:

$\begin{aligned}&-64{x^6}{y^{12}}+125{x^{16}}{y^3}\\&\Leftrightarrow-{\left(4 \right)^3}{\left( {{x^2}} \right)^3}{\left( {{y^4}} \right)^3} + {\left( 5 \right)^3}\left( x \right){\left( x \right)^{15}}{\left( y \right)^3} \\ & \Leftrightarrow {\left( {-4{x^2}{y^4}} \right)^3}+x{\left( 5 \right)^3}{\left( {{x^5}} \right)^3}{\left( y \right)^3}\\ &\Leftrightarrow {\left( {-4{x^2}{y^4}} \right)^3}+x{\left( {5{x^5}y} \right)^3} \\ \end{aligned}$

Thus, the option A is not sum of two cubic expressions.  

Step 2:  

For option B the expression $-32{x^6}{y^{12}}+125{x^{16}}{y^3}$ is simplified as follows:  

$\begin{aligned}&-32{x^6}{y^{12}} + 125{x^{16}}{y^3} \\ & \Leftrightarrow-\left( 4 \right){\left( 2 \right)^3}{\left( {{x^2}} \right)^3}{\left( {{y^4}} \right)^3} + {\left( 5 \right)^3}\left( x \right){\left( x \right)^{15}}{\left( y \right)^3} \\ & \Leftrightarrow4{\left( {-2{x^2}{y^4}} \right)^3} + x{\left( 5 \right)^3}{\left( {{x^5}} \right)^3}{\left( y \right)^3} \\ &\Leftrightarrow4{\left( {-2{x^2}{y^4}} \right)^3} + x{\left( {5{x^5}y} \right)^3} \\ \end{aligned}$

Thus, the option B is not sum of two cubic expressions.  

Step 3:  

For option C the expression $32{x^6}{y^{12}} + 125{x^9}{y^3}$ is simplified as follows:  

$\begin{aligned} &32{x^6}{y^{12}}+125{x^9}{y^3} \\ &\Leftrightarrow\left( 4 \right)\left( 8 \right){\left( {{x^2}} \right)^3}{\left( {{y^4}} \right)^3} + {\left( 5 \right)^3}{\left( {{x^3}} \right)^3}{\left( y \right)^3} \\ &\Leftrightarrow \left( 4 \right){\left( 2 \right)^3}{\left( {{x^2}} \right)^3}{\left( {{y^4}} \right)^3} + {\left( 5 \right)^3}{\left( {{x^3}} \right)^3}{\left( y \right)^3} \\ &\Leftrightarrow 4{\left( {2{x^2}{y^4}} \right)^3} + {\left( {5{x^3}y} \right)^3} \\ \end{aligned}$

Thus, the option C is not sum of two cubic expressions.  

Step 4:  

For option D the expression $64{x^6}{y^{12}}+125{x^9}{y^3}$ is simplified as follows:  

$\begin{aligned} &64{x^6}{y^{12}} + 125{x^9}{y^3} \\ &\Leftrightarrow {\left( 4 \right)^3}{\left( {{x^2}} \right)^3}{\left( {{y^4}} \right)^3} + {\left( 5 \right)^3}{\left( {{x^3}} \right)^3}{\left( y \right)^3} \\ &\Leftrightarrow {\left( {4{x^2}{y^4}} \right)^3} + {\left( {5{x^3}y} \right)^3} \\ \end{aligned}$

Here, the expression $64{x^6}{y^{12}} + 125{x^9}{y^3}$ is written as the sum of two cubic terms ${\left( {4{x^2}{y^4}} \right)^3}$and${\left( {5{x^3}y} \right)^3}$.  

Thus, the option D is expressed as the sum of two cubic expressions.  

Learn more:  

1. Which function has an inverse that is also a function? {(–1, –2), (0, 4), (1, 3), (5, 14), (7, 4)} {(–1, 2), (0, 4), (1, 5), (5, 4), (7, 2)} {(–1, 3), (0, 4), (1, 14), (5, 6), (7, 2)} {(–1, 4), (0, 4), (1, 2), (5, 3), (7, 1)}  

brainly.com/question/1632445  

2. A given line has the equation 10x + 2y = −2. what is the equation, in slope-intercept form, of the line that is parallel to the given line and passes through the point (0, 12)? y = ( )x + 12  

brainly.com/question/1473992  

3. what are the domain and range of the function f(x) = 3x + 5?  

brainly.com/question/3412497  

Answer Details :  

Grade: Middle School.  

Subject: Mathematics.  

Chapter: Linear equation.  

Keywords:  

Expression, cubes, quadratic, cubic expression, linear equation, zeros, 64x^6y^12, function, substitution, pets, direct substitution, elimination, graph, middle term factorization, fraction.