Assertion (A):

$\dfrac{8^{x+2}+8^{x+1}}{8^{x+2}-8^{x+1}} \\[1em] = \dfrac{x + 2 + x + 1}{x + 2 - x - 1} \\[1em] = 2x + 3$

Reason (R):

$\dfrac{8^{x+2}+8^{x+1}}{8^{x+2}-8^{x+1}} \\[1em] = \dfrac{8^{(x+2+x+1)}}{8^{(x+2-x-1)}} \\[1em] = \dfrac{x + 2 + x + 1}{x + 2 - x - 1} \\[1em] = 2x + 3$

1. A is true, R is false.
2. A is false, R is true.
3. Both A and R are true.
4. Both A and R are false.

Both A and R are false.

Explanation

$\dfrac{8^{x+2}+8^{x+1}}{8^{x+2}-8^{x+1}} = \dfrac{8^{x + 1} \times 8 + 8^{x + 1}}{8^{x + 1} \times 8 - 8^{x + 1}}\\[1em] = \dfrac{8^{x + 1}(8 + 1)}{8^{x + 1}(8 - 1)}\\[1em] = \dfrac{8^{x + 1} \times 9}{8^{x + 1} \times 7}\\[1em] = \dfrac{\cancel{8^{x + 1}} \times 9}{\cancel{8^{x + 1}} \times 7}\\[1em] = \dfrac{9}{7}$

∴ $\dfrac{8^{x+2}+8^{x+1}}{8^{x+2}-8^{x+1}}$ = $\dfrac{9}{7}$ ≠ 2x + 3

∴ Assertion (A) is false.

$\dfrac{8^{x+2}+8^{x+1}}{8^{x+2}-8^{x+1}}$ ≠ $\dfrac{8^{(x+2+x+1)}}{8^{(x+2-x-1)}}$

This step is incorrect because the powers of 8 cannot be directly combined in this way. The correct method is to factor out ${8^{x + 1}}$ as shown in assertion.

∴ Reason (R) is false.

Hence, both Assertion (A) and Reason (R) are false.