Ravi Kant Kumar

Find x, y, z and evaluate expression Given \(2^x \cdot 3^y \cdot 5^z = 2160\), find \(x, y, z\) and evaluate \(2^x \cdot 2^{-y} \cdot 5^{-z}\) Solution \[ 2160 = 2^4 \cdot 3^3 \cdot 5^1 \] \[ \Rightarrow x = 4,\; y = 3,\; z = 1 \] \[ 2^x \cdot 2^{-y} \cdot 5^{-z} \] […]

Solve √(a/b) = (b/a)^(1-2x) Solve: \(\sqrt{\frac{a}{b}} = \left(\frac{b}{a}\right)^{1-2x}\) Solution \[ \sqrt{\frac{a}{b}} = \left(\frac{b}{a}\right)^{1-2x} \] \[ \Rightarrow \left(\frac{a}{b}\right)^{1/2} = \left(\frac{a}{b}\right)^{-(1-2x)} \] \[ \Rightarrow \left(\frac{a}{b}\right)^{1/2} = \left(\frac{a}{b}\right)^{2x-1} \] \[ \Rightarrow \frac{1}{2} = 2x – 1 \] \[ \Rightarrow 2x = \frac{3}{2} \] \[ \Rightarrow x = \frac{3}{4} \] Final Answer: \[ \boxed{x = \frac{3}{4}} \] Next Question

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 \[ 4^{x-1} \times (0.5)^{3-2x} = \left(\frac{1}{8}\right)^x \] \[ \Rightarrow (2^2)^{x-1} \times (2^{-1})^{3-2x} = (2^{-3})^x \] \[ \Rightarrow 2^{2x-2} \times 2^{-3+2x} = 2^{-3x} \] \[ \Rightarrow 2^{2x-2-3+2x} = 2^{-3x} \] \[ \Rightarrow 2^{4x-5} = 2^{-3x} \] \[ \Rightarrow 4x – 5 = -3x

Solve system of exponential equations Solve: \[ 8^{x+1} = 16^{y+2}, \quad \left(\frac{1}{2}\right)^{3+x} = \left(\frac{1}{4}\right)^{3y} \] Solution \[ 8^{x+1} = 16^{y+2} \] \[ \Rightarrow (2^3)^{x+1} = (2^4)^{y+2} \] \[ \Rightarrow 2^{3x+3} = 2^{4y+8} \] \[ \Rightarrow 3x + 3 = 4y + 8 \quad …(1) \] \[ \left(\frac{1}{2}\right)^{3+x} = \left(\frac{1}{4}\right)^{3y} \] \[ \Rightarrow (2^{-1})^{3+x} = (2^{-2})^{3y}

Solve 3^(x-1) × 5^(2y-3) = 225 Solve: \(3^{x-1} \times 5^{2y-3} = 225\) Solution \[ 3^{x-1} \times 5^{2y-3} = 225 \] \[ \Rightarrow 3^{x-1} \times 5^{2y-3} = 3^2 \times 5^2 \] \[ \Rightarrow 3^{x-1} = 3^2,\quad 5^{2y-3} = 5^2 \] \[ \Rightarrow x-1 = 2 \] \[ \Rightarrow x = 3 \] \[ \Rightarrow 2y-3 =

Solve 4^(2x) = (cube root 16)^(-6/y) = (√8)^2 Solve: \(4^{2x} = (\sqrt[3]{16})^{-6/y} = (\sqrt{8})^2\) Solution \[ (\sqrt{8})^2 = (8^{1/2})^2 = 8 \] \[ \Rightarrow 4^{2x} = 8 \] \[ \Rightarrow (2^2)^{2x} = 2^3 \] \[ \Rightarrow 2^{4x} = 2^3 \] \[ \Rightarrow 4x = 3 \] \[ \Rightarrow x = \frac{3}{4} \] \[ (\sqrt[3]{16})^{-6/y} =

Solve 3^(x+1) = 27 × 3^4 Solve: \(3^{x+1} = 27 \times 3^4\) Solution \[ 3^{x+1} = 27 \times 3^4 \] \[ \Rightarrow 3^{x+1} = 3^3 \times 3^4 \] \[ \Rightarrow 3^{x+1} = 3^{7} \] \[ \Rightarrow x + 1 = 7 \] \[ \Rightarrow x = 6 \] Final Answer: \[ \boxed{x = 6} \]

Find x and y Find \(x\) and \(y\), if \(5^{3x} = 125\) and \(10^y = 0.001\) Solution \[ 5^{3x} = 125 \] \[ \Rightarrow 5^{3x} = 5^3 \] \[ \Rightarrow 3x = 3 \] \[ \Rightarrow x = 1 \] \[ 10^y = 0.001 \] \[ \Rightarrow 10^y = 10^{-3} \] \[ \Rightarrow y =

Find 2^(1+x) Find: \(2^{1+x}\), if \(3^{4x} = 81^{-1}\) and \(10^{1/y} = 0.0001\) Solution \[ 3^{4x} = 81^{-1} \] \[ \Rightarrow 3^{4x} = (3^4)^{-1} \] \[ \Rightarrow 3^{4x} = 3^{-4} \] \[ \Rightarrow 4x = -4 \] \[ \Rightarrow x = -1 \] \[ 10^{1/y} = 0.0001 \] \[ \Rightarrow 10^{1/y} = 10^{-4} \] \[ \Rightarrow

Find 2^(1+x) Find: \(2^{1+x}\), if \(3^{x+1} = 9^{x-2}\) Solution \[ 3^{x+1} = 9^{x-2} \] \[ \Rightarrow 3^{x+1} = (3^2)^{x-2} \] \[ \Rightarrow 3^{x+1} = 3^{2x-4} \] \[ \Rightarrow x+1 = 2x-4 \] \[ \Rightarrow x = 5 \] \[ 2^{1+x} = 2^{6} \] \[ = 64 \] Final Answer: \[ \boxed{64} \] Next Question /