Here is a transcription of the math work in the image, rewritten for clarity and proper structure:

Mercredi 09 Octobre 2024

La Correction du test diagnostique :

Exercice 1:

1. $A = \frac{3}{5} + \frac{2}{3}$
   $= \frac{3 \times 3}{5 \times 3} + \frac{2 \times 5}{3 \times 5}$
   $= \frac{9}{15} + \frac{10}{15}$
   $= \frac{19}{15}$

2. $B = \frac{3}{5} + \left( \frac{3}{5} \times \frac{4}{3} \right) - \frac{3}{2}$
   $= \frac{3}{5} + \frac{12}{15} - \frac{3}{2}$
   $= \frac{5 + 12}{15} - \frac{45}{30}$
   $= \frac{17}{15} - \frac{90}{30}$
   $= \frac{54}{30}$, and simplifications are detailed further.

3. $C = \frac{5^4 \times 5^3}{5^2}$
   $= 5^{4+3-2}$
   $= 5^{5}$

4. $D = 5 \big((n+1)(n-1) + 5 \big)$
   Expand and simplify:
   $= 5(n^2 - 1 + 5)$
   $= 5n^2 - 5$

5. $E = (n+1)(n^2 + n + 1)$
   $= n(n^2 + n + 1) + (n^2 + n + 1)$
   Combine terms:
   $= n^3 + n^2 + n + n^2 + n + 1$
   $= n^3 + 2n^2 + 2n + 1$

Identities and Properties:

1. $(a+b)(a-b) = a^2 - b^2$
2. $(a+b)^2 = a^2 + 2ab + b^2$
3. $(a-b)^2 = a^2 - 2ab + b^2$

Properties of exponents:

• $a^n \div a^m = a^{n-m}$
• $(a^m)^n = a^{m \times n}$
• $a^n \times a^m = a^{n+m}$

Simplifications of fractions:

$\frac{a}{b} + \frac{c}{d} = \frac{ad + cb}{bd}$
$\frac{a}{b} - \frac{c}{d} = \frac{ad - cb}{bd}$

Advanced problem:

$F = 4\sqrt{5} + 2\sqrt{45} - 4\sqrt{125}$

Simplify:

• $2\sqrt{45} = 6\sqrt{5}$
• $4\sqrt{125} = 20\sqrt{5}$

Combine terms:
$F = 4\sqrt{5} + 6\sqrt{5} - 20\sqrt{5}$
$= -10\sqrt{5}$

Verification: $-\sqrt{5}(12 + 1 - 16) = 0$.

Do you have any questions or would you like further details?

Follow-Up Questions:

1. Can you explain why $5^4 \times 5^3 = 5^7$?
2. How do you simplify $\frac{a}{b} + \frac{c}{d}$?
3. Why does $(a+b)(a-b) = a^2 - b^2$?
4. What is the significance of expanding $(n+1)(n^2 + n + 1)$?
5. How do we handle radicals like $4\sqrt{125}$?

Tip:

When working with exponents, always check the base. If they are the same, you can apply rules like addition or subtraction of powers directly.