If x¹⁰⁰ +  1/x¹⁰⁰ =3 then x⁵⁰⁰ + 1/x⁵⁰⁰ =?

We are given that:

\[
x^{100} + \frac{1}{x^{100}} = 3
\]

We need to find the value of:

\[
x^{500} + \frac{1}{x^{500}}
\]

### Step-by-Step Solution

Let \( y = x^{100} \). This simplifies the given equation to:

\[
y + \frac{1}{y} = 3
\]

Now, square both sides of this equation to express higher powers of \( y \):

\[
\left( y + \frac{1}{y} \right)^2 = 3^2
\]

This expands as:

\[
y^2 + 2 + \frac{1}{y^2} = 9
\]

Simplifying:

\[
y^2 + \frac{1}{y^2} = 9 - 2 = 7
\]

Next, cube both sides of \( y + \frac{1}{y} = 3 \) to express higher powers of \( y \):

\[
\left( y + \frac{1}{y} \right)^5 = (y^5 + \frac

If x¹⁰⁰ +  1/x¹⁰⁰ = 3, then x⁵⁰⁰ + 1/x⁵⁰⁰ =?

Discover how to solve the equation x¹⁰⁰ + 1/x¹⁰⁰ = 3 and find x⁵⁰⁰ + 1/x⁵⁰⁰ through a detailed step-by-step approach. This problem explores exponents, algebraic manipulations, and the use of polynomial identities. Suitable for high school students in Grades 9-12.

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