if x=(5)^1/3 +2 then the value of x^3-6x^2+12x-10 is

To solve for the expression \(x^3 - 6x^2 + 12x - 10\) given that \(x = (5)^{1/3} + 2\), we can start by simplifying this step by step. First, let’s compute \(x\) and then use that value to find the expression.

Calculating the Value of x

We have:

• \(x = (5)^{1/3} + 2\)

Here, \((5)^{1/3}\) represents the cube root of 5. Let’s denote \((5)^{1/3}\) as \(y\), which gives us:

• \(x = y + 2\)

Substituting x into the Expression

Now, let's compute \(x^3 - 6x^2 + 12x - 10\). We can first find \(x^3\), \(x^2\), and then substitute those values into the polynomial.

Finding x^2 and x^3

First, calculate \(x^2\):

• \(x^2 = (y + 2)^2 = y^2 + 4y + 4\)

Next, we calculate \(x^3\):

• \(x^3 = (y + 2)^3 = y^3 + 3y^2(2) + 3y(2^2) + 2^3\)
• \(x^3 = y^3 + 6y^2 + 12y + 8\)

Since \(y = (5)^{1/3}\), we have \(y^3 = 5\). Thus, we can substitute \(y^3\) into the equation:

• \(x^3 = 5 + 6y^2 + 12y + 8 = 13 + 6y^2 + 12y\)

Substituting to Find the Expression Value

Now we can plug \(x\), \(x^2\), and \(x^3\) back into the expression \(x^3 - 6x^2 + 12x - 10\):

• Substituting for \(x^3\): \(13 + 6y^2 + 12y - 6(y^2 + 4y + 4) + 12(y + 2) - 10\)

This simplifies to:

• \(= 13 + 6y^2 + 12y - 6y^2 - 24y - 24 + 12y + 24 - 10\)

Now, combine like terms:

• \(= 13 - 24 + 24 - 10 + (12y - 24y + 12y)\)
• \(= 13 - 10 = 3\)

Final Result

Hence, the value of the expression \(x^3 - 6x^2 + 12x - 10\) is \(3\). This result shows how we can systematically substitute and manipulate algebraic expressions to find the desired outcome.