We appreciate your visit to Which are sums of perfect cubes Check all that apply tex 8x 6 27 tex tex x 9 1 tex tex 81x 3 16x 6. 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!
Answer :
We want to determine which expressions are sums of perfect cubes. An expression is a sum of perfect cubes if each term can be written in the form [tex]$a^3$[/tex], where [tex]$a$[/tex] is an expression.
Below is the step-by-step check for each option:
1. [tex]$$8x^6 + 27$$[/tex]
- Notice that
[tex]$$8x^6 = (2x^2)^3$$[/tex]
because
[tex]$$(2x^2)^3 = 2^3 \cdot (x^2)^3 = 8x^6.$$[/tex]
- Also,
[tex]$$27 = 3^3.$$[/tex]
- Since both terms are perfect cubes, this expression is a sum of perfect cubes.
2. [tex]$$x^9 + 1$$[/tex]
- We can write
[tex]$$x^9 = (x^3)^3$$[/tex]
because
[tex]$$(x^3)^3 = x^9.$$[/tex]
- Also,
[tex]$$1 = 1^3.$$[/tex]
- Thus, this expression is also a sum of perfect cubes.
3. [tex]$$81x^3 + 16x^6$$[/tex]
- The coefficient [tex]$81$[/tex] is not a perfect cube (since [tex]$3^3 = 27$[/tex] and [tex]$4^3 = 64$[/tex], neither of which is [tex]$81$[/tex]).
- Similarly, the coefficient [tex]$16$[/tex] is not a perfect cube.
- Therefore, this expression is not a sum of perfect cubes.
4. [tex]$$x^6 + x^3$$[/tex]
- We have
[tex]$$x^6 = (x^2)^3$$[/tex]
because
[tex]$$(x^2)^3 = x^6.$$[/tex]
- And,
[tex]$$x^3 = (x)^3.$$[/tex]
- Since both terms are perfect cubes, this expression is a sum of perfect cubes.
5. [tex]$$27x^9 + x^{12}$$[/tex]
- For the first term:
[tex]$$27x^9 = (3x^3)^3$$[/tex]
because
[tex]$$(3x^3)^3 = 27x^9.$$[/tex]
- For the second term:
[tex]$$x^{12} = (x^4)^3$$[/tex]
since
[tex]$$(x^4)^3 = x^{12}.$$[/tex]
- Hence, this expression is a sum of perfect cubes.
6. [tex]$$9x^3 + 27x^9$$[/tex]
- Here, while
[tex]$$27x^9 = (3x^3)^3,$$[/tex]
the term [tex]$$9x^3$$[/tex] does not represent a perfect cube because [tex]$9$[/tex] is not a perfect cube (its cube root is not an integer).
- Thus, this expression is not a sum of perfect cubes.
Based on these computations, the expressions that are sums of perfect cubes are options 1, 2, 4, and 5.
Below is the step-by-step check for each option:
1. [tex]$$8x^6 + 27$$[/tex]
- Notice that
[tex]$$8x^6 = (2x^2)^3$$[/tex]
because
[tex]$$(2x^2)^3 = 2^3 \cdot (x^2)^3 = 8x^6.$$[/tex]
- Also,
[tex]$$27 = 3^3.$$[/tex]
- Since both terms are perfect cubes, this expression is a sum of perfect cubes.
2. [tex]$$x^9 + 1$$[/tex]
- We can write
[tex]$$x^9 = (x^3)^3$$[/tex]
because
[tex]$$(x^3)^3 = x^9.$$[/tex]
- Also,
[tex]$$1 = 1^3.$$[/tex]
- Thus, this expression is also a sum of perfect cubes.
3. [tex]$$81x^3 + 16x^6$$[/tex]
- The coefficient [tex]$81$[/tex] is not a perfect cube (since [tex]$3^3 = 27$[/tex] and [tex]$4^3 = 64$[/tex], neither of which is [tex]$81$[/tex]).
- Similarly, the coefficient [tex]$16$[/tex] is not a perfect cube.
- Therefore, this expression is not a sum of perfect cubes.
4. [tex]$$x^6 + x^3$$[/tex]
- We have
[tex]$$x^6 = (x^2)^3$$[/tex]
because
[tex]$$(x^2)^3 = x^6.$$[/tex]
- And,
[tex]$$x^3 = (x)^3.$$[/tex]
- Since both terms are perfect cubes, this expression is a sum of perfect cubes.
5. [tex]$$27x^9 + x^{12}$$[/tex]
- For the first term:
[tex]$$27x^9 = (3x^3)^3$$[/tex]
because
[tex]$$(3x^3)^3 = 27x^9.$$[/tex]
- For the second term:
[tex]$$x^{12} = (x^4)^3$$[/tex]
since
[tex]$$(x^4)^3 = x^{12}.$$[/tex]
- Hence, this expression is a sum of perfect cubes.
6. [tex]$$9x^3 + 27x^9$$[/tex]
- Here, while
[tex]$$27x^9 = (3x^3)^3,$$[/tex]
the term [tex]$$9x^3$$[/tex] does not represent a perfect cube because [tex]$9$[/tex] is not a perfect cube (its cube root is not an integer).
- Thus, this expression is not a sum of perfect cubes.
Based on these computations, the expressions that are sums of perfect cubes are options 1, 2, 4, and 5.
Thanks for taking the time to read Which are sums of perfect cubes Check all that apply tex 8x 6 27 tex tex x 9 1 tex tex 81x 3 16x 6. 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!
- Why do Businesses Exist Why does Starbucks Exist What Service does Starbucks Provide Really what is their product.
- The pattern of numbers below is an arithmetic sequence tex 14 24 34 44 54 ldots tex Which statement describes the recursive function used to..
- Morgan felt the need to streamline Edison Electric What changes did Morgan make.
Rewritten by : Barada
