Question

Prove the following:

`(x^("p"("q"-"r")))/(x^("q"("p"-"r"))) ÷ (x^"q"/x^"p")^"r"` = 1

Answer 1

L.H.S
= `(x^("p"("q"-"r")))/(x^("q"("p"-"r"))) ÷ (x^"q"/x^"p")^"r"`

= `(x^("p"("q"-"r")))/(x^("q"("p"-"r"))) ÷ x^"qr"/x^"pr"`   .....(Using (a^m)ⁿ = a^mn)

= `(x^("p"("q"-"r")))/(x^("q"("p"-"r"))) xx x^"pr"/x^"qr"`

= `(x^("pq"-"pr"))/x^("pq"- "qr") xx x^"pr"/x^"qr"`

= `(x^("pq"-"pr"+"pr"))/(x^("pq"-"qr"+"qr"`    .....(Using a^m x aⁿ = a^m+n)

= `(x^("pq"))/(x^("pq")`  
= 1 
= R.H.S
Hence proved.