May 2026

Find a, b, c in 1176 = 2^a × 3^b × 7^c Find \(a, b, c\) if \(1176 = 2^a \times 3^b \times 7^c\) Solution \[ 1176 = 2^3 \times 3^1 \times 7^2 \] \[ \text{Comparing with } 2^a \times 3^b \times 7^c \] \[ a = 3,\quad b = 1,\quad c = 2 \] […]

Find a, b, c in 49392 = a^4 b^2 c^3 Find \(a, b, c\) if \(49392 = a^4 b^2 c^3\) Solution \[ 49392 = 2^5 \times 3^2 \times 7^3 \] \[ = (2)^5 (3)^2 (7)^3 \] \[ \text{Compare with } a^4 b^2 c^3 \] \[ a = 2,\quad b = 3,\quad c = 7 \]

Solve 4^(x-1)(0.5)^(3-2x) = (1/8)^x Solve: \(4^{x-1}\times (0.5)^{3-2x} = \left(\frac{1}{8}\right)^x\) Solution \[ 4 = 2^2,\quad 0.5 = 2^{-1},\quad \frac{1}{8} = 2^{-3} \] \[ = (2^2)^{x-1} \cdot (2^{-1})^{3-2x} = (2^{-3})^x \] \[ = 2^{2x-2} \cdot 2^{-3+2x} = 2^{-3x} \] \[ = 2^{4x-5} = 2^{-3x} \] \[ \Rightarrow 4x – 5 = -3x \] \[ \Rightarrow 7x =

Solve 3^(2x+4) + 1 = 2·3^(x+2) Solve: \(3^{2x+4} + 1 = 2\cdot3^{x+2}\) Solution \[ 3^{2x+4} + 1 = 2\cdot3^{x+2} \] \[ \text{Let } t = 3^{x+2} \] \[ \Rightarrow t^2 + 1 = 2t \] \[ \Rightarrow t^2 – 2t + 1 = 0 \] \[ \Rightarrow (t – 1)^2 = 0 \] \[ \Rightarrow

Solve 2^(2x) – 2^(x+3) + 2^4 = 0 Solve: \(2^{2x} – 2^{x+3} + 2^4 = 0\) Solution \[ 2^{2x} – 2^{x+3} + 16 = 0 \] \[ \text{Let } 2^x = t \] \[ \Rightarrow t^2 – 8t + 16 = 0 \] \[ \Rightarrow (t – 4)^2 = 0 \] \[ \Rightarrow t =

Simplify exponential expression Simplify: \[ \frac{6\cdot8^{n+1} + 16\cdot2^{3n-2}}{10\cdot2^{3n-1} – 7\cdot8^n} \] Solution \[ 8 = 2^3 \] \[ = \frac{6\cdot(2^3)^{n+1} + 16\cdot2^{3n-2}}{10\cdot2^{3n-1} – 7\cdot(2^3)^n} \] \[ = \frac{6\cdot2^{3n+3} + 2^4\cdot2^{3n-2}}{10\cdot2^{3n-1} – 7\cdot2^{3n}} \] \[ = \frac{2^{3n+2}(12 + 1)}{2^{3n-1}(10 – 14)} \] \[ = \frac{13\cdot2^{3n+2}}{-4\cdot2^{3n-1}} \] \[ = -13\cdot2^3 \] \[ = -104 \] Final Answer:

Simplify given exponential expression Simplify: \[ \frac{5\cdot25^{n+1}-25\cdot5^{2n}}{5\cdot5^{2n+3}-25^{n+1}} \] Solution \[ 25 = 5^2 \] \[ = \frac{5\cdot(5^2)^{n+1} – 5^2\cdot5^{2n}}{5\cdot5^{2n+3} – (5^2)^{n+1}} \] \[ = \frac{5^{2n+3} – 5^{2n+2}}{5^{2n+4} – 5^{2n+2}} \] \[ = \frac{5^{2n+2}(5 – 1)}{5^{2n+2}(25 – 1)} \] \[ = \frac{4}{24} = \frac{1}{6} \] Final Answer: \[ \boxed{\frac{1}{6}} \] Next Question / Full Exercise

Simplify (3^n × 9^(n+1)) / (3^(n-1) × 9^(n-1)) Simplify: \(\frac{3^n \times 9^{n+1}}{3^{n-1} \times 9^{n-1}}\) Solution \[ = \frac{3^n \cdot (3^2)^{n+1}}{3^{n-1} \cdot (3^2)^{n-1}} \] \[ = \frac{3^n \cdot 3^{2n+2}}{3^{n-1} \cdot 3^{2n-2}} \] \[ = 3^{\,n+2n+2-(n-1+2n-2)} \] \[ = 3^{\,3n+2-(3n-3)} = 3^5 \] \[ = 243 \] Final Answer: \[ \boxed{243} \] Next Question / Full Exercise

Proof of exponent identity Prove: \[ \left(\frac{x^a}{x^b}\right)^{a^2+ab+b^2} \times \left(\frac{x^b}{x^c}\right)^{b^2+bc+c^2} \times \left(\frac{x^c}{x^a}\right)^{c^2+ca+a^2} = 1 \] Proof \[ = x^{(a-b)(a^2+ab+b^2)} \cdot x^{(b-c)(b^2+bc+c^2)} \cdot x^{(c-a)(c^2+ca+a^2)} \] \[ = x^{(a^3-b^3) + (b^3-c^3) + (c^3-a^3)} \] \[ = x^{0} \] \[ = 1 \] Hence Proved Next Question / Full Exercise

Proof of (a^-1 + b^-1)^-1 = ab/(a+b) Prove: \((a^{-1}+b^{-1})^{-1} = \frac{ab}{a+b}\) Proof \[ (a^{-1}+b^{-1})^{-1} \] \[ = \left(\frac{1}{a} + \frac{1}{b}\right)^{-1} \] \[ = \left(\frac{a+b}{ab}\right)^{-1} \] \[ = \frac{ab}{a+b} \] Hence Proved Next Question / Full Exercise