Educational

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

Proof of (a+b+c)/(a^-1 + b^-1 + c^-1) = abc Prove: \(\frac{a+b+c}{a^{-1}+b^{-1}+c^{-1}} = abc\) Proof \[ = \frac{a+b+c}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}} \] \[ = \frac{a+b+c}{\frac{bc+ca+ab}{abc}} \] \[ = (a+b+c)\cdot \frac{abc}{ab+bc+ca} \] \[ = abc \] Hence Proved Next Question / Full Exercise

Solve 2^(3x-7) = 256 Solve: \(2^{3x-7} = 256\) Solution \[ 2^{3x-7} = 256 \] \[ 256 = 2^8 \] \[ \Rightarrow 2^{3x-7} = 2^8 \] \[ \Rightarrow 3x – 7 = 8 \] \[ \Rightarrow 3x = 15 \] \[ \Rightarrow x = 5 \] Final Answer: \[ \boxed{x = 5} \] Next Question /

Solve 4^(2x) = 1/32 Solve: \(4^{2x} = \frac{1}{32}\) Solution \[ 4^{2x} = \frac{1}{32} \] \[ (2^2)^{2x} = 2^{-5} \] \[ 2^{4x} = 2^{-5} \] \[ \Rightarrow 4x = -5 \] \[ \Rightarrow x = -\frac{5}{4} \] Final Answer: \[ \boxed{x = -\frac{5}{4}} \] Next Question / Full Exercise

Solve 2^(5x+3) = 8^(x+3) Solve: \(2^{5x+3} = 8^{x+3}\) Solution \[ 2^{5x+3} = (2^3)^{x+3} \] \[ = 2^{5x+3} = 2^{3x+9} \] \[ \Rightarrow 5x + 3 = 3x + 9 \] \[ \Rightarrow 2x = 6 \] \[ \Rightarrow x = 3 \] Final Answer: \[ \boxed{x = 3} \] Next Question / Full Exercise

Solve 2^(x+1) = 4^(x-3) Solve: \(2^{x+1} = 4^{x-3}\) Solution \[ 2^{x+1} = (2^2)^{x-3} \] \[ = 2^{x+1} = 2^{2x-6} \] \[ \Rightarrow x + 1 = 2x – 6 \] \[ \Rightarrow x = 7 \] Final Answer: \[ \boxed{x = 7} \] Next Question / Full Exercise

Solve 7^(2x+3) = 1 Solve: \(7^{2x+3} = 1\) Solution \[ 7^{2x+3} = 1 \] \[ \text{Since } 7^0 = 1 \] \[ \Rightarrow 2x + 3 = 0 \] \[ \Rightarrow 2x = -3 \] \[ \Rightarrow x = -\frac{3}{2} \] Final Answer: \[ \boxed{x = -\frac{3}{2}} \] Next Question / Full Exercise

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

Simplify 3(a^4b^3)^10 × 5(a^2b^2)^3 Simplify: \(3(a^4b^3)^{10} \times 5(a^2b^2)^3\) Solution \[ = 3 \times 5 \cdot (a^4b^3)^{10} \cdot (a^2b^2)^3 \] \[ = 15 \cdot a^{40}b^{30} \cdot a^{6}b^{6} \] \[ = 15 \cdot a^{40+6} \cdot b^{30+6} \] \[ = 15a^{46}b^{36} \] Final Answer: \[ \boxed{15a^{46}b^{36}} \] Next Question / Full Exercise