Question

please solve this sum on exponents.​

Answer 1

Answer:

√[ x^( p - q ) } × √{ x^( q - r ) } × √{ x^( r - p ) } = 1

Step-by-step explanation:

= > √[ x^( p - q ) } × √{ x^( q - r ) } × √{ x^( r - p ) }

Few properties / formulae related this topic :

• √a = a^( 1 / 2 )
• a^( n ) × a^( m ) = a^( n + m )
• ( a^n )^m = a^( nm )
• a^0 = 1 { a ≠ 0 }

= > √[ x^( p - q ) } × √{ x^( q - r ) } × √{ x^( r - p ) }

= > ( x^{ p - q } )^( 1 / 2 ) × ( x^{ q - r } )^( 1 / 2 ) × ( x^{ r - p } )^( 1 / 2 ) { Using √a = a^( 1 / 2 ) }

= > x^{ ( p - q ) × 1 / 2 } × x^{ ( q - r ) × 1 / 2 } × x^{ ( r - p ) × 1 / 2 } ( Using ( a^n )^m = a^( nm ) }

= > x^{ ( p - q ) / 2 } × x^{ ( q - r ) / 2 } × x^{ ( r - p ) / 2 }

= > x^{ ( p - q ) / 2 + ( q - r ) / 2 + ( r - p ) / 2 } { Using a^n × a^m = a^( n + m ) }

= > x^{ ( p - q + q - r + r - p ) / 2 }

= > x^( 0 / 2 )

= > x^( 0 )

= > 1 { Using a^0 = 1 , a ≠ 0 }

Hence proved.