Educational

Solve 13^(√x) = 4^4 – 3^4 – 6 Solve: \(13^{\sqrt{x}} = 4^4 – 3^4 – 6\) Solution \[ 13^{\sqrt{x}} = 4^4 – 3^4 – 6 \] \[ \Rightarrow 13^{\sqrt{x}} = 256 – 81 – 6 \] \[ \Rightarrow 13^{\sqrt{x}} = 169 \] \[ \Rightarrow 13^{\sqrt{x}} = 13^2 \] \[ \Rightarrow \sqrt{x} = 2 \] \[ […]

Solve 5^(2x+3) = 1 Solve: \(5^{2x+3} = 1\) Solution \[ 5^{2x+3} = 1 \] \[ \Rightarrow 1 = 5^0 \] \[ \Rightarrow 5^{2x+3} = 5^0 \] \[ \Rightarrow 2x + 3 = 0 \] \[ \Rightarrow 2x = -3 \] \[ \Rightarrow x = -\frac{3}{2} \] Final Answer: \[ \boxed{x = -\frac{3}{2}} \] Next Question

Solve (cube root of 4)^(2x+1/2) = 1/32 Solve: \((\sqrt[3]{4})^{2x+\frac{1}{2}} = \frac{1}{32}\) Solution \[ (\sqrt[3]{4})^{2x+\frac{1}{2}} = \frac{1}{32} \] \[ \Rightarrow (4^{1/3})^{2x+\frac{1}{2}} = 2^{-5} \] \[ \Rightarrow 4^{\frac{2x+\frac{1}{2}}{3}} = 2^{-5} \] \[ \Rightarrow (2^2)^{\frac{2x+\frac{1}{2}}{3}} = 2^{-5} \] \[ \Rightarrow 2^{\frac{2(2x+\frac{1}{2})}{3}} = 2^{-5} \] \[ \Rightarrow 2^{\frac{4x+1}{3}} = 2^{-5} \] \[ \Rightarrow \frac{4x+1}{3} = -5 \] \[ \Rightarrow

Solve 2^(x-7) × 5^(x-4) = 1250 Solve: \(2^{x-7} \times 5^{x-4} = 1250\) Solution \[ 2^{x-7} \times 5^{x-4} = 1250 \] \[ \Rightarrow 2^{x-7} \times 5^{x-4} = 2 \times 5^4 \] \[ \Rightarrow \frac{2^x}{2^7} \times \frac{5^x}{5^4} = 2 \times 5^4 \] \[ \Rightarrow \frac{2^x \cdot 5^x}{2^7 \cdot 5^4} = 2 \cdot 5^4 \] \[ \Rightarrow 2^x

Solve 5^(x-2) × 3^(2x-3) = 135 Solve: \(5^{x-2} \times 3^{2x-3} = 135\) Solution \[ 5^{x-2} \times 3^{2x-3} = 135 \] \[ \Rightarrow 5^{x-2} \times 3^{2x-3} = 5 \times 3^3 \] \[ \Rightarrow \frac{5^x}{5^2} \times \frac{3^{2x}}{3^3} = 5 \times 3^3 \] \[ \Rightarrow \frac{5^x \cdot 3^{2x}}{5^2 \cdot 3^3} = 5 \cdot 3^3 \] \[ \Rightarrow 5^x

Solve (3/5)^x (5/3)^2x = 125/27 Solve: \(\left(\frac{3}{5}\right)^x \left(\frac{5}{3}\right)^{2x} = \frac{125}{27}\) Solution \[ \left(\frac{3}{5}\right)^x \left(\frac{5}{3}\right)^{2x} = \frac{125}{27} \] \[ \Rightarrow \frac{3^x}{5^x} \cdot \frac{5^{2x}}{3^{2x}} = \frac{125}{27} \] \[ \Rightarrow \frac{3^x \cdot 5^{2x}}{5^x \cdot 3^{2x}} = \frac{125}{27} \] \[ \Rightarrow \frac{5^x}{3^x} = \frac{125}{27} \] \[ \Rightarrow \left(\frac{5}{3}\right)^x = \left(\frac{5}{3}\right)^3 \] \[ \Rightarrow x = 3 \] Final Answer:

Solve (2^3)^4 = (2^2)^x Solve: \((2^3)^4 = (2^2)^x\) Solution \[ (2^3)^4 = (2^2)^x \] \[ = 2^{12} = 2^{2x} \] \[ \Rightarrow 12 = 2x \] \[ \Rightarrow x = 6 \] Final Answer: \[ \boxed{x = 6} \] Next Question / Full Exercise

Solve 2^(5x)/2^x = 5th root of 2^20 Solve: \(\frac{2^{5x}}{2^x} = \sqrt[5]{2^{20}}\) Solution \[ \frac{2^{5x}}{2^x} = \sqrt[5]{2^{20}} \] \[ = 2^{5x-x} = (2^{20})^{1/5} \] \[ = 2^{4x} = 2^4 \] \[ \Rightarrow 4x = 4 \] \[ \Rightarrow x = 1 \] Final Answer: \[ \boxed{x = 1} \] Next Question / Full Exercise

Solve 27^x = 9 / 3^x Solve: \(27^x = \frac{9}{3^x}\) Solution \[ 27 = 3^3,\quad 9 = 3^2 \] \[ (3^3)^x = \frac{3^2}{3^x} \] \[ 3^{3x} = 3^{2-x} \] \[ \Rightarrow 3x = 2 – x \] \[ \Rightarrow 4x = 2 \] \[ \Rightarrow x = \frac{1}{2} \] Final Answer: \[ \boxed{x = \frac{1}{2}}

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