1Fill in the table: the value of \( 7^x \) for \( x = -3, -2, -1, 0, 1, 2, 3 \).
2Fill in the table: the value of \( 5^{-x} \) for \( x = -3, -2, -1, 0, 1, 2, 3 \).
3Solve and examine: \( 3^x = 9 \)
4Solve and examine: \( 5^{x-1} = 25 \)
5Solve and examine: \( \frac{1}{5^{2x-4}} = 125 \)
6Solve and examine: \( 4^{x-2} = 0.125 \)
7Solve and examine: \( \left(\frac{3}{5}\right)^x = \left(1\frac{2}{3}\right)^3 \)
8Solve and examine: \( 2^x \times 3^{x+1} = 18 \)
9Solve: \( 4^{\frac{1-x}{1+x}} = 4^{\frac{1}{3}} \)
10Solve: \( \sqrt[2x+4]{4^{x+8}} = \sqrt[6]{128} \)
11Solve: \( 2^{x+1} + 2^{x+2} + 2^{x+3} = 448 \)
12Solve: \( 3^{x+1} - 3^x = 162 \)
13Solve: \( 4^{x+1} - 8 \times 4^{x-1} = 32 \)
14Solve: \( 4 \times 3^{x+1} - 3^{x+2} - 3^{x-1} = 72 \)
15Solve: \( 3^{x+2} + 3^{x+1} + 2 \times 3^x = 126 \)
16Solve: \( 2^x + 3^{x-2} = 3^x - 2^{x+1} \)
17Solve: \( 8^{x-1} - 2^{3x-2} + 8 = 0 \)
18Solve: \( \left(\frac{1}{4}\right)^{2-\sqrt{5x+1}} = 4 \times 2^{\sqrt{5x+1}} \)
19Solve: \( 5^x + \frac{1}{5^x} = 5\frac{1}{5} \)
20Solve: \( 7^x + \frac{1}{7^x} = 7\frac{1}{7} \)
21Solve: \( 9^x + \frac{1}{9^x} = 9\frac{1}{9} \)
22Solve: \( 4^x + \frac{1}{4^x} = 16\frac{1}{16} \)
23Solve: \( 5^x + 5^{-x} = 25\frac{1}{25} \)
24Solve: \( 81 \times 3^x + 3^{-x} = 30 \)
25Solve: \( 4 \times 3^{x+1} - 9^x = 27 \)
26Solve: \( 3 \times 2^{p+1} - 4^p = 8 \)
27Solve: \( 5^{2x} - 6 \times 5^{x+1} + 125 = 0 \)
28Solve: \( 2^{x-2} + 2^{3-x} = 3 \)
29Solve: \( 5^{x+1} + 5^{2-x} = 126 \)
30Solve: \( 3^{2y} - 4 \times 3^y + 3 = 0 \)
31Solve \( 16^x - 5 \times 4^{x+1} + 64 = 0 \). Prove that the value of x satisfies \( 5^x + \frac{125}{5^x} = 30 \).
32If \( x = 3^{\frac{1}{3}} + 3^{-\frac{1}{3}} \), prove that \( 3x(x^2 - 3) = 10 \).
33If \( x = 2^{\frac{1}{3}} - 2^{-\frac{1}{3}} \), prove that \( 2x^3 + 6x - 3 = 0 \).
34Harisaran has two rectangular fields, each of area 360 m². The difference between length and breadth is 2 m for the first and 9 m for the second. Find the length and breadth of both fields.
35Does it take the same length of wire to fence both fields?
36With wire costing Rs. 10 per metre, which field costs more to fence, and by how much?
37A footpath of equal width surrounds a 16 m by 12 m grassland, and the total area is 320 m². Taking the path's width as x, write the equation.
38Find the width of the footpath.
39Simplify: \( \frac{1}{a - b} - \frac{2b}{a^2 - b^2} \)
40Simplify: \( \frac{a - 4}{a^2 - 4a + 16} + \frac{a + 4}{a^2 + 4a + 16} + \frac{128}{a^4 + 16a^2 + 256} \)
41Simplify: \( \frac{2a - 6}{a^2 - 9a + 20} - \frac{a - 1}{a^2 - 7a + 12} - \frac{a - 2}{a^2 - 8a + 15} \)
42Simplify: \( \frac{a + b}{2ab}(a + b - c) + \frac{b + c}{2bc}(b + c - a) + \frac{c + a}{2ac}(c + a - b) \)
43Solve: \( 3^{x+2} + 3^{2-x} = 82 \)
44Solve: \( \frac{3^{2x} + 1}{3^x} = \frac{82}{9} \)
45If \( x = 1 + 2^{\frac{1}{3}} + 2^{\frac{2}{3}} \), prove that \( x(x^2 - 3x - 3) = 1 \).
46If \( x = 3 + 3^{\frac{1}{3}} + 3^{\frac{2}{3}} \), prove that \( x(x^2 - 9x + 18) = 12 \).
47If \( x = 2 - 2^{\frac{1}{3}} + 2^{\frac{2}{3}} \), prove that \( x(x^2 - 6x + 18) = 22 \).
48Why can the indices be equated in \( 3^{x-1} = 3^4 \) but not directly in \( 3^{x+1} + 3^x = 3^4 \)?
49Solving with \( 3^x = a \) gave \( a = -3 \) and \( a = 9 \). What is x?
50Solve: \( 9^x = 27 \)