please-prove-1-x-x-1-

Question Number 213725 by zakirullah last updated on 14/Nov/24

p l e a s e p r o v e \frac{1}{x} = x^{- 1}

Answered by Rasheed.Sindhi last updated on 14/Nov/24

\frac{1}{x} = \frac{x^{0}}{x^{1}} = x^{0 - 1} = x^{- 1}

Answered by Rasheed.Sindhi last updated on 14/Nov/24

x^{- 1} = x^{0 - 1} = \frac{x^{0}}{x^{1}} = \frac{1}{x}

Commented by zakirullah last updated on 15/Nov/24

v e r y t r u e b u t s o m e s t u d e n t s a s k e d a b o u t

t h e p o w e r o r e x p o n e n t t h a t w h y t h e y c h a n g e i t s

s i g n w h e n i t g o f r o m n o m e r a t o r t o d e n o m i n a t o r a n d

f r o m d e n o m i n a t o r t o n o m e r a t o r ?

p l e a s e g i v e m e a n y s u g g i s t i o n .

Commented by Rasheed.Sindhi last updated on 15/Nov/24

(1)

a^{0} = 1 (d e f i n i t i o n)

\frac{1}{a} = \frac{1 . a^{- 1}}{a . a^{- 1}}

(n u m e r a t o r a n d d e n o m i n a t o r m a y

b e m u l t i p l i e d b y s a m e n u m b e r)

= \frac{a^{- 1}}{a^{1 + (- 1)}} = \frac{a^{- 1}}{a^{0}} = \frac{a^{- 1}}{1} = a^{- 1}

(2)

{\begin{cases} a^{x} \times a^{y} = a^{x + y} \\ \frac{a^{x}}{a^{y}} = a^{x - y} (*) \end{cases} (A s R u l e s)

1 = \frac{a}{a} = a^{1 - 1} = a^{0} \Rightarrow a^{0} = 1

N o w,

\frac{1}{a^{n}} = \frac{a^{0}}{a^{n}} = a^{0 - n} = a^{- n}

(*) a^{x} \times a^{y} = a^{x + y} \Rightarrow a^{x} = \frac{a^{x + y}}{a^{y}} = a^{(x + y) - y}

\Rightarrow \frac{a^{m}}{a^{n}} = a^{m - n}

Answered by Ghisom last updated on 14/Nov/24

we define the symbol a^{n}

a^{n} := \underset{n times}{\underset{⏟}{a \times a \times a \times \dots}}

a \times b = c \Leftrightarrow a = b \div c

\Rightarrow a^{m} \times a^{n} = a^{m + n} \Leftrightarrow a^{m + n} \div a^{n} = a^{m}

this can be written as

a^{m + n} \div a^{n} = a^{(m + n) - n}

generally

a^{m} \div a^{n} = a^{m - n}

what happens when m = n ?

a^{m} \div a^{m} = a^{m - m} = a^{0}

but a^{m} \div a^{m} = 1 \Rightarrow a^{0} = 1

what happens when m < n ?

let n = m + k

a^{m} \div a^{m + k} = a^{m - (m + k)} = a^{- k}

an example :

2^{3} \div 2^{5} = 2^{- 2} (?)

2^{3} \div 2^{5} = 8 \div 32 = \frac{1}{4} = \frac{1}{2^{2}}

\Rightarrow 2^{- 2} = \frac{1}{2^{2}}

\Rightarrow

a^{- m} = \frac{1}{a^{m}}

Commented by zakirullah last updated on 15/Nov/24

w e l l - d o n e s i r G