1. The problem involves simplifying and understanding the expressions arranged in a pyramid shape. 2. Let's analyze each expression step-by-step. 3. Top row: The number is simply $5$. 4. Second row: - Left: $\frac{1}{24}x^3$ means $\frac{1}{24}$ times $x^3$. - Right: $2x^2 + \frac{2}{x} + 1$ is a sum of three terms: $2x^2$, $\frac{2}{x}$, and $1$. 5. Third row: - Left: $\frac{2x}{2x + 5}$ is a fraction with numerator $2x$ and denominator $2x + 5$. - Middle: $\frac{1}{2}$ is a constant fraction. - Right: $\frac{8x}{3(x + 2)}$ is a fraction with numerator $8x$ and denominator $3(x + 2)$. 6. Fourth row: - Left: $\frac{x^4 + 5x^3}{3}$ means the sum $x^4 + 5x^3$ divided by $3$. - Second: $\frac{x^2 - y^2}{7}$ is the difference of squares $x^2 - y^2$ divided by $7$. - Third: $\frac{1}{x} + 1$ is the sum of $\frac{1}{x}$ and $1$. - Right: $\frac{9x^2}{8}$ is $9x^2$ divided by $8$. 7. Important rules: - When dividing polynomials or terms, divide numerator and denominator separately. - Factor expressions when possible, e.g., $x^2 - y^2 = (x - y)(x + y)$. - Simplify fractions by canceling common factors. 8. Simplifications: - $\frac{1}{24}x^3$ stays as is. - $2x^2 + \frac{2}{x} + 1$ cannot be combined further without common denominators. - $\frac{2x}{2x + 5}$ stays as is; numerator and denominator share no common factors. - $\frac{8x}{3(x + 2)}$ stays as is. - $\frac{x^4 + 5x^3}{3} = \frac{x^3(x + 5)}{3}$ factoring out $x^3$. - $\frac{x^2 - y^2}{7} = \frac{(x - y)(x + y)}{7}$ using difference of squares. - $\frac{1}{x} + 1$ can be written as $\frac{1 + x}{x}$ if combined over common denominator. - $\frac{9x^2}{8}$ stays as is. 9. Final expressions: - $5$ - $\frac{1}{24}x^3$ - $2x^2 + \frac{2}{x} + 1$ - $\frac{2x}{2x + 5}$ - $\frac{1}{2}$ - $\frac{8x}{3(x + 2)}$ - $\frac{x^3(x + 5)}{3}$ - $\frac{(x - y)(x + y)}{7}$ - $\frac{1 + x}{x}$ - $\frac{9x^2}{8}$ This completes the analysis and simplification of all expressions in the pyramid.