Problem 63
Question
Factor using the formula for the sum or difference of two cubes. $$64 x^{3}+27$$
Step-by-Step Solution
Verified Answer
Upon factoring the expression \(64x^3 + 27\) using the sum of cubes formula, it simplifies to \((4x + 3) (16x^2 - 12x + 9)\).
1Step 1: Identify the Cubes
Recognize the given expression as \(64x^3 + 27\) which can be written as \((4x)^3 + 3^3\).
2Step 2: Apply Sum of Cubes Formula
Apply the sum of cubes formula, which is \((a^3 + b^3) = (a+b)(a^2 - ab + b^2)\). Here, \(a = 4x\) and \(b = 3\). So, \((4x)^3 + 3^3 = (4x+3)( (4x)^2 - (4x*3) + 3^2)\).
3Step 3: Simplify the Result
Simplify the above expression to get the final answer. This would be \((4x + 3) (16x^2 - 12x + 9)\).
Other exercises in this chapter
Problem 62
Simplify each exponential expression. $$ \left(\frac{-30 a^{14} b^{8}}{10 a^{17} b^{-2}}\right)^{3} $$
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simplify each complex rational expression. $$ \frac{\frac{1}{x}+\frac{1}{y}}{x+y} $$
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Evaluate each expression in Exercises \(55-66,\) or indicate that the root is not a real number. $$ \sqrt[5]{(-3)^{5}} $$
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In Exercises 59–66, perform the indicated operations. Indicate the degree of the resulting polynomial. $$ \left(x^{3}+7 x y-5 y^{2}\right)-\left(6 x^{3}-x y+4 y
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