Question

please find me this answer from Monday I have exams please please please please please please please please please please please please please please please please please please please please please​

Answer 1

$\huge{ \underline{ \overline{ \mid{ \mathfrak{ \purple{ \: \: QUESTION \: \: } \mid}}}}}$

$\mathsf{ \: \:Prove \: \: that, \: \: } \\ \\ \mathtt{ (a {}^{ \frac{1}{x - y} } ) {}^{ \frac{1}{x - z} } \times (a {}^{ \frac{1}{y - z} } ) {}^{ \frac{1}{y - x} } \times (a {}^{ \frac{1}{z - x} } ) {}^{\frac{1}{z - y}} = 1 }$

$\huge{ \underline{ \overline{ \mid{ \mathfrak{ \purple{ \: \: ANSWER \: \: } \mid}}}}}$

$\mathtt{L.H.S. = (a {}^{ \frac{1}{x - y} } ) {}^{ \frac{1}{x - z} } \times (a {}^{ \frac{1}{y - z} } ) {}^{ \frac{1}{y - x} } \times (a {}^{ \frac{1}{z - x} } ) {}^{\frac{1}{z - y}} } \\ \\ \mathtt{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = (a) {}^{ \frac{1}{x - y} . \frac{1}{x - z} } \times (a) {}^{ \frac{1}{y - z} . \frac{1}{y - x} } \times (a) {}^{ \frac{1}{z - x} .\frac{1}{z - y} } }\\ \\ \mathtt{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = (a) {}^{ \frac{1}{(x - y)(x - z)} } \times (a) {}^{ \frac{1}{(y - z)(y - x)} } \times (a) {}^{ \frac{1}{(z - x)(z - y)} }} \\ \\ \mathtt{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: =( a) {}^{\frac{1}{(x - y).(x - z)} +\frac{1}{(y - z).(y - x)} + \frac{1}{(z - x).(z - y)} }} \\ \\ \mathtt{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: =( a) {}^{\frac{1}{( - 1).(x - y).( z - x)} +\frac{1}{( - 1).(y - z).(x - y)} + \frac{1}{( - 1).(z - x).(y - z )} }} \\ \\ \mathtt{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: =( a) {}^{\frac{( - 1)}{(x - y).(z - x)} +\frac{( - 1)}{(y - z).(x - y)} + \frac{( - 1)}{(z - x).(y - z )} }} \\ \\ \mathtt{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = (a) {}^{ \frac{( - 1).(y - z) + ( - 1).(z - x) + ( - 1).(x - y)}{(x - y)(y - z)(z - x)} }} \\ \\ \mathtt{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = (a) {}^{ \frac{ - y + z - z + x - x + y}{(x - y)(y - z)(z - x)}}} \\ \\ \mathtt{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = (a) {}^{ \frac{0}{(x - y)(y - z)(z - x)}} } \\ \\ \mathtt{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = (a) {}^{0}} \\ \\ \mathtt{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = 1 \: \: = R.H.S. }$

FORMULA USED :-

$\mathtt{1. \: (p {}^{ \frac{1}{m} } ) {}^{ \frac{1}{n} } = p {}^{ \frac{1}{mn} } }\\ \\ \mathtt{2. \: p {}^{m} \times p{}^{n} = p{}^{m + n}} \\ \\ \mathtt{3. \: p - q = ( - 1).(q - p)} \\ \\ \mathtt{4. \: p {}^{ \frac{0}{m} } = p {}^{0}} \\ \\ \mathtt{5. \: p {}^{0 = 1} }$