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Question
Prove the following:
`(x^("p"("q"-"r")))/(x^("q"("p"-"r"))) ÷ (x^"q"/x^"p")^"r"` = 1
Solution
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 (am)n = amn)
= `(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 am x an = am+n)
= `(x^("pq"))/(x^("pq")`
= 1
= R.H.S
Hence proved.
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