College

We appreciate your visit to Which are sums of perfect cubes Check all that apply A tex 8x 6 27 tex B tex x 9 1 tex C tex 81x. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!

Which are sums of perfect cubes? Check all that apply.

A. [tex]8x^6 + 27[/tex]

B. [tex]x^9 + 1[/tex]

C. [tex]81x^3 + 16x^6[/tex]

D. [tex]x^6 + x^3[/tex]

E. [tex]27x^9 + x^{12}[/tex]

F. [tex]9x^3 + 27x^9[/tex]

Answer :

We want to express each term in the form of a perfect cube, i.e., as something like [tex]$(\text{expression})^3$[/tex]. Let’s check each option one by one.

[tex]$$\textbf{Option 1: } 8x^6 + 27$$[/tex]

- The term [tex]$8x^6$[/tex] can be written as
[tex]$$8x^6 = (2x^2)^3,$$[/tex]
because [tex]$(2x^2)^3 = 2^3 \cdot (x^2)^3 = 8x^6.$[/tex]
- The term [tex]$27$[/tex] can be written as
[tex]$$27 = 3^3.$$[/tex]

Since both terms are perfect cubes, option 1 is a sum of perfect cubes.

[tex]$$\textbf{Option 2: } x^9 + 1$$[/tex]

- The term [tex]$x^9$[/tex] can be expressed as
[tex]$$x^9 = (x^3)^3,$$[/tex]
because [tex]$(x^3)^3 = x^{3\cdot3} = x^9.$[/tex]
- The term [tex]$1$[/tex] is
[tex]$$1 = 1^3.$$[/tex]

Thus, option 2 is a sum of perfect cubes.

[tex]$$\textbf{Option 3: } 81x^3 + 16x^6$$[/tex]

- Although [tex]$x^6$[/tex] can be written as [tex]$(x^2)^3$[/tex], notice that the coefficient [tex]$16$[/tex] is not a perfect cube.
- Similarly, [tex]$81$[/tex] is not a perfect cube.

Therefore, option 3 is not a sum of perfect cubes.

[tex]$$\textbf{Option 4: } x^6 + x^3$$[/tex]

- The term [tex]$x^6$[/tex] can be written as
[tex]$$x^6 = (x^2)^3.$$[/tex]
- The term [tex]$x^3$[/tex] is
[tex]$$x^3 = (x)^3.$$[/tex]

Thus, option 4 is a sum of perfect cubes.

[tex]$$\textbf{Option 5: } 27x^9 + x^{12}$$[/tex]

- The term [tex]$27x^9$[/tex] can be written as
[tex]$$27x^9 = (3x^3)^3,$$[/tex]
because [tex]$(3x^3)^3 = 3^3 \cdot (x^3)^3 = 27x^9.$[/tex]
- The term [tex]$x^{12}$[/tex] can be rewritten as
[tex]$$x^{12} = (x^4)^3.$$[/tex]

So, option 5 is a sum of perfect cubes.

[tex]$$\textbf{Option 6: } 9x^3 + 27x^9$$[/tex]

- The term [tex]$27x^9$[/tex] can be written as
[tex]$$27x^9 = (3x^3)^3.$$[/tex]
- However, the term [tex]$9x^3$[/tex] cannot be written as a perfect cube because [tex]$9$[/tex] is not a perfect cube.

Thus, option 6 is not a sum of perfect cubes.

In summary, the expressions that are sums of perfect cubes are:

[tex]$$\boxed{1,\ 2,\ 4,\ 5}$$[/tex]

Thanks for taking the time to read Which are sums of perfect cubes Check all that apply A tex 8x 6 27 tex B tex x 9 1 tex C tex 81x. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada