Factor a^(5n)+a^(2n)
usually we factor out the power with the smaller exponent, so
a^(5n) + a^(2n)
= a^(2n) (a^(3n) + 1)
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To factor the expression a^(5n) + a^(2n), we can first observe that both terms have a common factor of a^(2n). We can then rewrite the expression as:
a^(2n) * (a^(3n) + 1)
Now, let's focus on factoring the expression within the parentheses (a^(3n) + 1). This can be done by recognizing it as a sum of cubes. Recall the sum of cubes formula:
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
We can apply this formula by considering a^(3n) as a cube and 1 as the second term:
a^(3n) + 1 = (a^n)^3 + 1^3
Using the sum of cubes formula, we can factor it as:
(a^n + 1)(a^(2n) - a^n + 1)
Therefore, the factored form of the expression a^(5n) + a^(2n) is:
a^(2n) * (a^n + 1)(a^(2n) - a^n + 1)