Ravi Kant Kumar

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

Proof of (x^a/x^b)^c (x^b/x^c)^a (x^c/x^a)^b = 1 Prove: \(\left(\frac{x^a}{x^b}\right)^c \times \left(\frac{x^b}{x^c}\right)^a \times \left(\frac{x^c}{x^a}\right)^b = 1\) Proof \[ = \left(x^{a-b}\right)^c \cdot \left(x^{b-c}\right)^a \cdot \left(x^{c-a}\right)^b \] \[ = x^{(a-b)c} \cdot x^{(b-c)a} \cdot x^{(c-a)b} \] \[ = x^{ac – bc + ab – ac + bc – ab} \] \[ = x^0 \] \[ = 1 \] Hence

Find (a + b)^ab when a = 3, b = -2 Find: \((a + b)^{ab}\), where \(a = 3, b = -2\) Solution \[ = (3 – 2)^{-6} \] \[ = 1^{-6} \] \[ = 1 \] Final Answer: \[ \boxed{1} \] Next Question / Full Exercise

Find a^b + b^a when a = 3, b = -2 Find: \(a^b + b^a\), where \(a = 3, b = -2\) Solution \[ = 3^{-2} + (-2)^3 \] \[ = \frac{1}{9} – 8 \] \[ = -\frac{71}{9} \] Final Answer: \[ \boxed{-\frac{71}{9}} \] Next Question / Full Exercise

Find a^a + b^b when a = 3, b = -2 Find: \(a^a + b^b\), where \(a = 3, b = -2\) Solution \[ = 3^3 + (-2)^{-2} \] \[ = 27 + \frac{1}{4} \] \[ = \frac{109}{4} \] Final Answer: \[ \boxed{\frac{109}{4}} \] Next Question / Full Exercise

Simplify (a^(3n-9))^6 / a^(2n-4) Simplify: \(\frac{(a^{3n-9})^6}{a^{2n-4}}\) Solution \[ = \frac{a^{18n-54}}{a^{2n-4}} \] \[ = a^{16n-50} \] Final Answer: \[ \boxed{a^{16n-50}} \] Next Question / Full Exercise

Simplify ((x^2y^2)/a^2b^3)^n Simplify: \(\left(\frac{x^2y^2}{a^2b^3}\right)^n\) Solution \[ = \frac{x^{2n}y^{2n}}{a^{2n}b^{3n}} \] Final Answer: \[ \boxed{\frac{x^{2n}y^{2n}}{a^{2n}b^{3n}} } \] Next Question / Full Exercise

Simplify (4ab^2(-5ab^3))/10a^2b^2 Simplify: \(\frac{4ab^2(-5ab^3)}{10a^2b^2}\) Solution \[ = \frac{-20a^2b^5}{10a^2b^2} \] \[ = -2b^3 \] Final Answer: \[ \boxed{-2b^3} \] Next Question / Full Exercise