https://file.aiquickdraw.com/mathbot/file/1726985729479oz3wprz8.jpg

The expression in the image is:

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

To expand this expression, we need to apply the binomial expansion (or cube the expression directly). Let's break it down:

We have:
\[
\left( a + b \right)^3 = a^3 + 3a^2b + 3ab^2 + b^3
\]
where \( a = x^2 \) and \( b = \frac{1}{x^2} \).

Now, let's apply this step-by-step.

1. \( a^3 = (x^2)^3 = x^6 \)
2. \( b^3 = \left(\frac{1}{x^2}\right)^3 = \frac{1}{x^6} \)
3. \( 3a^2b = 3(x^2)^2 \cdot \frac{1}{x^2} = 3x^4 \)
4. \( 3ab^2 = 3x^2 \cdot \left(\frac{1}{x^2}\right)^2 = 3 \cdot \frac{1}{x^4} \)

Putting it all together, we get:

\[
x^6 + 3x^4 + 3\frac{1}{x^4} + \frac{1}{x^6}
\]

Thus, the expanded form of the expression is:

\[
x^6 + 3x^4 + 3\frac{1}{x^4} + \frac{1}{x^6}
\]

Would you like a more detailed explanation of any step or further assistance with this? Here are some related questions to explore:

1. What is the general form of the binomial expansion?
2. How can we simplify higher powers of rational expressions?
3. What are the applications of binomial expansions in calculus?
4. How can you apply this technique to more complex algebraic expressions?
5. What happens if we cube an expression with a negative term?

**Tip:** Always look out for symmetry in expressions like this one; it can make the expansion much quicker!

Expand and simplify the expression (x^2 + 1/x^2)^3 using the binomial expansion formula. This step-by-step solution includes key algebraic concepts like powers, exponents, and binomial expansions. Suitable for students in Grades 10-12.

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