May 2026

Proof of Given Relation Question \[ 3^x = 5^y = 75^z \] Solution \[ \text{Let } 3^x = 5^y = 75^z = k \] \[ 3^x = k \Rightarrow x = \log_3 k,\quad 5^y = k \Rightarrow y = \log_5 k \] \[ 75^z = k \Rightarrow (3 \cdot 5^2)^z = k \] \[ 3^z […]

Proof of Given Relation Question \[ a^x = b^y = c^z,\quad b^2 = ac \] Solution \[ \text{Let } a^x = b^y = c^z = k \] \[ a = k^{1/x},\quad b = k^{1/y},\quad c = k^{1/z} \] \[ b^2 = ac \Rightarrow (k^{1/y})^2 = k^{1/x} \cdot k^{1/z} \] \[ k^{2/y} = k^{1/x + 1/z}

Proof of Given Relation Question \[ 2^x = 3^y = 6^{-z} \] Solution Let \(2^x = 3^y = 6^{-z} = k\) \[ x = \log_2 k,\quad y = \log_3 k,\quad -z = \log_6 k \] \[ \frac{1}{x} = \log_k 2,\quad \frac{1}{y} = \log_k 3,\quad \frac{1}{z} = -\log_k 6 \] \[ \frac{1}{x} + \frac{1}{y} + \frac{1}{z}

Proof of Given Relation Question \[ 2^x = x^y = 12^z \] Solution Let \(2^x = x^y = 12^z = k\) \[ 2^x = k \Rightarrow x = \log_2 k \] \[ x^y = k \Rightarrow y = \frac{\log k}{\log x} \] \[ 12^z = k \Rightarrow z = \log_{12} k \] \[ \frac{1}{z} =

Proof of Given Expression Question \[ (x^{a^2+b^2}/x^{2ab})^{a+b}(x^{b^2+c^2}/x^{2bc})^{b+c}(x^{c^2+a^2}/x^{2ca})^{c+a} \] Solution \[ = (x^{(a-b)^2})^{a+b}(x^{(b-c)^2})^{b+c}(x^{(c-a)^2})^{c+a} \] \[ = x^{(a-b)^2(a+b)} \cdot x^{(b-c)^2(b+c)} \cdot x^{(c-a)^2(c+a)} \] \[ = x^{(a^3-a^2b-ab^2+b^3) + (b^3-b^2c-bc^2+c^3) + (c^3-c^2a-ca^2+a^3)} \] \[ = x^{2(a^3+b^3+c^3) – (a^2b+ab^2+b^2c+bc^2+c^2a+ca^2)} \] \[ = x^{2(a^3+b^3+c^3)} \] Answer \[ \boxed{x^{2(a^3+b^3+c^3)}} \] Next Question / Full Exercise

Proof of Given Expression = 1 Question \[ (a^{x+1}/a^{y+1})^{x+y}(a^{y+2}/a^{z+2})^{y+z}(a^{z+3}/a^{x+3})^{z+x} \] Solution \[ = (a^{x-y})^{x+y}(a^{y-z})^{y+z}(a^{z-x})^{z+x} \] \[ = a^{(x-y)(x+y)} \cdot a^{(y-z)(y+z)} \cdot a^{(z-x)(z+x)} \] \[ = a^{(x^2-y^2)+(y^2-z^2)+(z^2-x^2)} \] \[ = a^0 \] \[ = 1 \] Answer \[ \boxed{1} \] Next Question / Full Exercise

Proof of Given Expression = x Question \[ \left\{(x^{a-1})^{\frac{1}{a-1}}\right\}^{\frac{a}{a+1}} \] Solution \[ = (x^{a-1})^{\frac{1}{a-1} \cdot \frac{a}{a+1}} \] \[ = x^{\frac{a}{a+1}} \] \[ = x \] Answer \[ \boxed{x} \] Next Question / Full Exercise

Proof of Given Expression = 1 Question \[ (1/x^{a-b})^{\frac{1}{a-c}} (1/x^{b-c})^{\frac{1}{b-a}} (1/x^{c-a})^{\frac{1}{c-b}} \] Solution \[ = x^{-\frac{a-b}{a-c}} \cdot x^{-\frac{b-c}{b-a}} \cdot x^{-\frac{c-a}{c-b}} \] \[ = x^{-\left[\frac{a-b}{a-c} + \frac{b-c}{b-a} + \frac{c-a}{c-b}\right]} \] \[ = x^{-(1+1+1-2)} = x^{0} \] \[ = 1 \] Answer \[ \boxed{1} \] Next Question / Full Exercise

Proof of Given Expression = 1 Question \[ \left[ \left(\frac{x^{a(a-b)}}{x^{a(a+b)}}\right) \div \left(\frac{x^{b(b-a)}}{x^{b(b+a)}}\right) \right]^{a+b} \] Solution \[ = \left[ \frac{x^{a(a-b)-a(a+b)}} {x^{b(b-a)-b(b+a)}} \right]^{a+b} \] \[ = \left[ \frac{x^{-2ab}}{x^{-2ab}} \right]^{a+b} \] \[ = (1)^{a+b} \] \[ = 1 \] Answer \[ \boxed{1} \] Next Question / Full Exercise

Proof of 1/(1+x^(a-b)) + 1/(1+x^(b-a)) = 1 Question \[ \frac{1}{1+x^{a-b}} + \frac{1}{1+x^{b-a}} \] Solution \[ = \frac{1}{1+\frac{x^a}{x^b}} + \frac{1}{1+\frac{x^b}{x^a}} \] \[ = \frac{1}{\frac{x^b + x^a}{x^b}} + \frac{1}{\frac{x^a + x^b}{x^a}} \] \[ = \frac{x^b}{x^a + x^b} + \frac{x^a}{x^a + x^b} \] \[ = \frac{x^b + x^a}{x^a + x^b} \] \[ = 1 \] Answer \[ \boxed{1}