Factor 64+125a^(3) completely. Answer:

Recognize as sum of cubes: Recognize the expression as a sum of two cubes. The given expression is 64 + 125a^(3). We can rewrite 64 as (4)^3 and 125a^(3) as (5a)^3. This allows us to express the given sum as a sum of cubes: (4)^3 + (5a)^3.Apply sum of cubes formula: Apply the sum of cubes formula. The sum of cubes formula is a^3 + b^3 = (a + b)(a^2 - ab + b^2). We will apply this formula to our expression where a = 4 and b = 5a.Substitute values: Substitute the values into the sum of cubes formula. Using the values a = 4 and b = 5a, we get the factorization (4 + 5a)((4)^2 - (4)(5a) + (5a)^2).Simplify factors: Simplify the factors. Now we simplify each factor: First factor: (4 + 5a) remains the same. Second factor: (4)^2 - (4)(5a) + (5a)^2 simplifies to 16 - 20a + 25a^2.Write final factorization: Write the final factorization. The complete factorization of the expression is (4 + 5a)(16 - 20a + 25a^2).

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Math Problems

Grade 8

Estimate positive square roots

Full solution

Q. Factor

64+125 a^{3}

completely.

\newline

Answer:

Recognize as sum of cubes: Recognize the expression as a sum of two cubes. The given expression is $64 + 125a^{3}$ . We can rewrite $64$ as $(4)^{3}$ and $125a^{3}$ as $(5a)^{3}$ . This allows us to express the given sum as a sum of cubes: $(4)^{3} + (5a)^{3}$ .
Apply sum of cubes formula: Apply the sum of cubes formula. $\newline$ The sum of cubes formula is $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ . We will apply this formula to our expression where $a = 4$ and $b = 5a$ .
Substitute values: Substitute the values into the sum of cubes formula. $\newline$ Using the values $a = 4$ and $b = 5a$ , we get the factorization $(4 + 5a)((4)^2 - (4)(5a) + (5a)^2)$ .
Simplify factors: Simplify the factors. $\newline$ Now we simplify each factor: $\newline$ First factor: $(4 + 5a)$ remains the same. $\newline$ Second factor: $(4)^2 - (4)(5a) + (5a)^2$ simplifies to $16 - 20a + 25a^2$ .
Write final factorization: Write the final factorization. $\newline$ The complete factorization of the expression is $(4 + 5a)(16 - 20a + 25a^2)$ .