Factor completely: $\newline$ $a^{6}-64 v^{6}$ $\newline$ Answer:

View step-by-step help

Home

Math Problems

Grade 7

Solve equations with variable exponents

Full solution

Q. Factor completely:

\newline

a^{6}-64 v^{6}

\newline

Answer:

Recognize as difference of squares: Recognize the expression as a difference of two squares. $\newline$ The expression $a^{6} - 64v^{6}$ can be written as $(a^{3})^2 - (8v^{3})^2$ , which is a difference of two squares since $(a^{3})^2$ is the square of $a^{3}$ and $(8v^{3})^2$ is the square of $8v^{3}$ .
Apply squares formula: Apply the difference of squares formula. $\newline$ The difference of squares formula is $a^2 - b^2 = (a + b)(a - b)$ . Here, $a = a^{3}$ and $b = 8v^{3}$ . So, we can factor the expression as follows: $\newline$ $(a^{3} + 8v^{3})(a^{3} - 8v^{3})$
Recognize as cubes: Recognize that both factors are differences and sums of cubes. $\newline$ Both $a^{3} + 8v^{3}$ and $a^{3} - 8v^{3}$ can be further factored because they are sums and differences of cubes, respectively. The sum of cubes formula is $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ , and the difference of cubes formula is $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ .
Apply sum of cubes formula: Apply the sum of cubes formula to the first factor. $\newline$ Using the sum of cubes formula on $a^{3} + 8v^{3}$ , we get: $\newline$ $a^{3} + 8v^{3} = a^{3} + (2v)^{3} = (a + 2v)(a^{2} - a(2v) + (2v)^{2})$ $\newline$ Simplifying, we get: $\newline$ $(a + 2v)(a^{2} - 2av + 4v^{2})$
Apply difference of cubes formula: Apply the difference of cubes formula to the second factor. $\newline$ Using the difference of cubes formula on $a^{3} - 8v^{3}$ , we get: $\newline$ $a^{3} - 8v^{3} = a^{3} - (2v)^{3} = (a - 2v)(a^2 + a(2v) + (2v)^2)$ $\newline$ Simplifying, we get: $\newline$ $(a - 2v)(a^2 + 2av + 4v^2)$
Combine factored forms: Combine the factored forms of both factors. $\newline$ Now we combine the factored forms from Step $4$ and Step $5$ to get the completely factored form of the original expression: $\newline$ $(a + 2v)(a^2 - 2av + 4v^2)(a - 2v)(a^2 + 2av + 4v^2)$