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  1. প্রমাণ কর যে, \(\left(a^{\tfrac{m}{n}}\right)^p = a^{\tfrac{mp}{n}}\), যেখানে \(m, p \in \mathbb{Z}\) এবং \(n \in \mathbb{N}\)।
  2. প্রমাণ কর যে, \(\left(a^{\tfrac{1}{m}}\right)^{\tfrac{1}{n}} = a^{\tfrac{1}{mn}}\), যেখানে \(m, n \in \mathbb{Z}, m \ne 0, n \ne 0\)।
  3. প্রমাণ কর যে, \(\left(ab\right)^{\tfrac{m}{n}} = \left(a^{\tfrac{m}{n}}\right)\!\left(b^{\tfrac{m}{n}}\right)\), যেখানে \(m \in \mathbb{Z}, n \in \mathbb{N}\)।
  4. দেখাও যে —
    1. (ক) \(\big(a^{\tfrac{1}{3}} - b^{\tfrac{1}{3}}\big) \big(a^{\tfrac{2}{3}} + a^{\tfrac{1}{3}}b^{\tfrac{1}{3}} + b^{\tfrac{2}{3}}\big) = a - b\)
    2. (খ) \( \dfrac{a^3+a^{-3}+ 1} {a^{\frac{3}{2}}+a^{\frac{-3}{2}}+ 1} = a^{\frac{3}{2}}+a^{\frac{-3}{2}}-1 \)
  5. সরল কর —
    1. (ক)  \( \dfrac{ \left(\dfrac{a+b}{b}\right)^{\frac{a}{a-b}} \times \left(\dfrac{a-b}{a}\right)^{\frac{a}{a-b}} } { \left(\dfrac{a+b}{b}\right)^{\frac{b}{a-b}} \times \left(\dfrac{a-b}{a}\right)^{\frac{b}{a-b}} } \)
    2. (খ) \(\dfrac{a^{\frac{3}{2}} + ab}{ab - b^3} - \dfrac{\sqrt{a}}{\sqrt{a} - b}\)
    3. (গ) \( \left\{ \left(x^{\frac{1}{a}}\right)^{\tfrac{a^2 - b^2}{a - b}} \right\}^{\!\tfrac{a}{a + b}} \)
    4. (ঘ) \(\dfrac{1}{1 + a^{-m}b^{n} + a^{-m}c^{p}} + \dfrac{1}{1 + b^{-n}c^{p} + b^{-n}a^{m}} + \dfrac{1}{1 + c^{-p}a^{m} + c^{-p}b^{n}}\)
    5. (ঙ) \( \sqrt[bc]{\dfrac{x^{\frac{b}{c}}}{x^\frac{c}{b}}} \times \sqrt[ca]{\dfrac{x^{\frac{c}{a}}}{x^\frac{c}{a}}} \times \sqrt[ab]{\dfrac{x^{\frac{a}{b}}}{x^\frac{a}{b}}} \)
    6. (চ) \(\dfrac{(a^2 - b^{-2})^a(a - b^{-1})^{b - a}} {(b^2 - a^{-2})^b(b + a^{-1})^{a - b}}\)
  6. দেখাও যে —

    (ক) যদি \(x = a^{q + r} b^{p},\; y = a^{r + p} b^{q},\; z = a^{p + q} b^{r}\) হয়,  তবে \(x^{q - r} \cdot y^{r - p} \cdot z^{p - q} = 1\)। 

    (খ) যদি \(a^p = b,\; b^q = c\) এবং \(c^r = a\) হয়,  তবে \(pqr = 1\)। 

    (গ) যদি \(a^x = p,\; a^y = q\) এবং \(a^z = (p^y q^x)^z\) হয়,  তবে \(xyz = 1\)।
  7. (ক) যদি \(x\sqrt[3]{a} + y\sqrt[3]{b} + z\sqrt[3]{c} = 0\) এবং \(a^2 = bc\) হয়,  তবে দেখাও যে, \(a x^3 + b y^3 + c z^3 = 3a xyz\)। 

    (খ) যদি \(x = (\!a + b\!)^{\tfrac{1}{3}} + (\!a - b\!)^{\tfrac{1}{3}}\) এবং \(a^2 - b^2 = c^3\) হয়,  তবে দেখাও যে, \(x^3 - 3cx - 2a = 0\)। 

    (গ) যদি \(a = 2^{\tfrac{1}{3}} + 2^{-\tfrac{1}{3}}\) হয়,  তবে দেখাও যে \(2a^3 - 6a = 5\)। 

    (ঘ) যদি \(a^2 + 2 = 3^{\tfrac{2}{3}} + 3^{-\tfrac{2}{3}}\) এবং \(a \ge 0\) হয়,  তবে দেখাও যে \(3a^3 + 9a = 8\)। 

    (ঙ) যদি \(a^2 = b^3\) হয়,  তবে দেখাও যে \(\left(\dfrac{a}{b}\right)^{\tfrac{3}{2}} + \left(\dfrac{b}{a}\right)^{\tfrac{2}{3}} = a^{\tfrac{1}{2}} + b^{-\tfrac{1}{3}}\)। 

    (চ) যদি \(b = 1 + 3^{\tfrac{2}{3}} + 3^{\tfrac{1}{3}}\) হয়,  তবে দেখাও যে \(b^3 - 3b^2 - 6b - 4 = 0\)।

    (ছ) যদি  \( \ a+ b+c= 0 \ \) হয়,  তবে দেখাও যে  \( \dfrac{1}{x^b+x^{-c}+1}+ \dfrac{1}{x^c+x^{-a}+1}+ \dfrac{1}{x^a+x^{-b}+1} = 1 \)
  8. (ক) যদি \(a^x = b,\; b^y = c\) এবং \(c^z = 1\) হয়,  তবে \(xyz\) এর মান নির্ণয় কর। 

    (খ) যদি \(x^a = y^b = z^c\) এবং \(xyz = 1\) হয়,  তবে \(ab + bc + ca\) এর মান নির্ণয় কর। 

    (গ) যদি \(9^x = 27^y\) হয়,  তবে \(\dfrac{x}{y}\) এর মান নির্ণয় কর।
  9. সমাধান কর —
    1. (ক) \(3^{2x + 2} + 27^{x + 1} = 36\)
    2. (খ) \(5^x + 3^y = 8,\; 5^{x - 1} + 3^{y - 1} = 2\)
    3. (গ) \(4^{3y - 2} = 16^{x + y},\; 3^{x + 2y} = 9^{2x + 1}\)
    4. (ঘ) \(2^{2x + 1} \cdot 2^{3y + 1} = 8,\; 2^{x + 2} \cdot 2^{y + 2} = 16\)
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