Question

r) \\(\frac{3^{-2} - 3^{-3}}{3^{-2} + 3^{-3}}\\)

Explanation:

Step1: Rewrite negative exponents

Recall $a^{-n}=\frac{1}{a^n}$, so:
$3^{-2}=\frac{1}{3^2}=\frac{1}{9}$, $3^{-3}=\frac{1}{3^3}=\frac{1}{27}$
Substitute into the expression:
$\frac{\frac{1}{9} - \frac{1}{27}}{\frac{1}{9} + \frac{1}{27}}$

Step2: Find common denominator (27)

Convert fractions to have denominator 27:
$\frac{1}{9}=\frac{3}{27}$, so:
$\frac{\frac{3}{27} - \frac{1}{27}}{\frac{3}{27} + \frac{1}{27}}$

Step3: Simplify numerator and denominator

Calculate numerator: $\frac{3-1}{27}=\frac{2}{27}$
Calculate denominator: $\frac{3+1}{27}=\frac{4}{27}$
Expression becomes: $\frac{\frac{2}{27}}{\frac{4}{27}}$

Step4: Divide the fractions

Dividing by a fraction is multiplying by its reciprocal:
$\frac{2}{27} \times \frac{27}{4} = \frac{2}{4}$

Step5: Simplify the final fraction

$\frac{2}{4}=\frac{1}{2}$

Answer:

$\frac{1}{2}$