Question

plz solve question no i​

Answer 1

Formula Applied :

$\underbrace{\sf Laws\ of\ Exponents}:\\\\ \sf 1.\ a^x\times a^y=a^{x+y}\\\\\sf 2.\ \dfrac{a^x}{a^y}=a^{x-y}\\\\\sf 3.\ (a^m)^n=a^{mn}\\\\\sf 4.\ \sqrt[\sf{n}]{\sf{x}}=x^{\frac{1}{n}}$

Solution :

$\sf \sqrt[\sf x+y]{\dfrac{\sf a^{x^2}}{\sf a^{y^2}}} \times \sqrt[\sf y+z]{\dfrac{\sf a^{y^2}}{\sf a^{z^2}}}\times \sqrt[\sf z+x]{\dfrac{\sf a^{z^2}}{\sf a^{x^2}}}$

$\to \sf \left( \dfrac{a^{x^2}}{a^{y^2}}\right)^{\frac{1}{x+y}}\times \left(\dfrac{a^{y^2}}{a^{z^2}}\right)^{\frac{1}{y+z}}\times \left(\dfrac{a^{z^2}}{a^{x^2}}\right)^{\frac{1}{z+x}}$

Apply 2nd law of exponent ,

$\to \sf \left(a^{x^2-y^2}\right)^{\frac{1}{x+y}}\times \left(a^{y^2-z^2}\right)^{\frac{1}{y+z}}\times \left(a^{z^2-x^2}\right)^{\frac{1}{z+x}}$

Apply 3rd law of exponent ,

$\to \sf a^{\frac{x^2-y^2}{x+y}}\times a^{\frac{y^2-z^2}{y+z}}\times a^{\frac{z^2-x^2}{z+x}}$

Apply a² - b² = ( a+b )( a-b ) ,

$\to \sf a^{\frac{(x+y)(x-y)}{x+y}}\times a^{\frac{(y+z)(y-z)}{y+z}}\times a^{\frac{(z+x)(z-x)}{z+x}}$

$\to \sf a^{x-y}\times a^{y-z}\times a^{z-x}$

Apply 1st law of exponent ,

$\to \sf a^{[(x-y)+(y-z)+(z-x)]}$

$\to \sf a^{x-y+y-z+z-x}$

$\to \sf a^{\circ}$

$\to \sf \red{1}\ \; \bigstar$

Answer 2

Formula Applied :

$\begin{gathered}\underbrace{\sf Laws\ of\ Exponents}:\\\\ \sf 1.\ a^x\times a^y=a^{x+y}\\\\\sf 2.\ \dfrac{a^x}{a^y}=a^{x-y}\\\\\sf 3.\ (a^m)^n=a^{mn}\\\\\sf 4.\ \sqrt[\sf{n}]{\sf{x}}=x^{\frac{1}{n}}\end{gathered}$

Solution :

$\sf \sqrt[\sf x+y]{\dfrac{\sf a^{x^2}}{\sf a^{y^2}}} \times \sqrt[\sf y+z]{\dfrac{\sf a^{y^2}}{\sf a^{z^2}}}\times \sqrt[\sf z+x]{\dfrac{\sf a^{z^2}}{\sf a^{x^2}}} </p><p>x+y$

$\to \sf ( \dfrac{a^{x^2}}{a^{y^2}})^{\frac{1}{x+y}}\times (\dfrac{a^{y^2}}{a^{z^2}})^{\frac{1}{y+z}}\times (\dfrac{a^{z^2}}{a^{x^2}})^{\frac{1}{z+x}}$

Apply 2nd law of exponent ,

$\to \sf (a^{x^2-y^2})^{\frac{1}{x+y}}\times (a^{y^2-z^2})^{\frac{1}{y+z}}\times (a^{z^2-x^2})^{\frac{1}{z+x}}$

Apply 3rd law of exponent ,

$\to \sf a^{\frac{x^2-y^2}{x+y}}\times a^{\frac{y^2-z^2}{y+z}}\times a^{\frac{z^2-x^2}{z+x}}$

Apply a² - b² = ( a+b )( a-b ) ,

$\to \sf a^{\frac{(x+y)(x-y)}{x+y}}\times a^{\frac{(y+z)(y-z)}{y+z}}\times a^{\frac{(z+x)(z-x)}{z+x}}$

Apply 1st law of exponent ,

$\to \sf a^{[(x-y)+(y-z)+(z-x)]}$

$\to \sf a^{x-y+y-z+z-x}$

$\to \sf a^{\circ}</p><p>$

$\to \sf \red{1}\ \; \bigstar$