Question

Factor: 27x^3-64step by step please

Answer 1

Solve this problem using the difference of cubes formula, since both terms are perfect cubes.

According to the difference of cubes formula, given a³ - b³ = (a - b)(a² + ab+ b²).

So, follow the steps to solve this problem.

Step 01: Find "a".

Comparing with the equation above:

`a^3=27x^3`

Take the cubic root from both sides:

`\begin{gathered} \sqrt[3]{a^3}=\sqrt[3]{27x^3} \\ a^{(3)/(3)=}\sqrt[3]{27}\cdot\sqrt[3]{x^3} \end{gathered}`

Factoring 27 = 3 * 3 *3 = 3³. Then,

`\begin{gathered} a=\sqrt[3]{3^3}\cdot\sqrt[3]{x^3} \\ a=3x \end{gathered}`

Step 01: Find "b".

`b^3=64`

Taking the cubic root from both sides and factoring 64:

64 = 4*4*4. Then,

`\begin{gathered} \sqrt[3]{b^3}=\sqrt[3]{4^3} \\ b=4 \end{gathered}`

Step 03: Substitute "a" and "b" in the formula.

a³ - b³ = (a - b)(a² + ab+ b²).

`\begin{gathered} 27x^3-64=(3x-4)\cdot\lbrack(3x)^2+3x\cdot4+4^2\rbrack \\ =(3x-4)\cdot(9x^2+12x+16) \end{gathered}`

Answer:

`27x^3-64=(3x-4)\cdot(9x^2+12x+16)`