Simplify- i xa-b-xc a-b xb-c-xa b-c xc-a-xb c-a-ii-xl-xm-xm-xn-xn-xl

(i) ( x^a+b/x^c )^a b (x^b+c/x^a )^b c ( x^c+a/x^b )^c a =(x^a+b c)^a b.(x^b+c a)^b c.(x^c+a b)^c a =x^(a b)(a+b c).x^(b c)(b+c a).x^(c a)(c+a b) =x^a^2 b^2 ac+ ...

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Byju's Answer

Standard IX

Mathematics

Laws of Exponents for Real Numbers

Simplify: i x...

Question

Simplify: $(i) {(\frac{x^{a + b}}{x^{c}})}^{a - b} {(\frac{x^{b + c}}{x^{a}})}^{b - c} {(\frac{x^{c + a}}{x^{b}})}^{c - a} (i i) \sqrt[l m]{\frac{x^{l}}{x^{m}}} \times \sqrt[m n]{\frac{x^{m}}{x^{n}}} \times \sqrt[n l]{\frac{x^{n}}{x^{l}}}$

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Solution

$(i) {(\frac{x^{a + b}}{x^{c}})}^{a - b} {(\frac{x^{b + c}}{x^{a}})}^{b - c} {(\frac{x^{c + a}}{x^{b}})}^{c - a} = (x^{a + b - c})^{a - b} . (x^{b + c - a})^{b - c} . (x^{c + a - b})^{c - a} = x^{(a - b) (a + b - c)} . x^{(b - c) (b + c - a)} . x^{(c - a) (c + a - b)} = x^{a^{2} - b^{2} - a c + b c} . x^{b^{2} - c^{2} - a b + a c} . x^{c^{2} - a^{2} - b c + a b} = x^{a^{2} - b^{2} - a c + b c + b^{2} - c^{2} - a b + a c + c^{2} - a^{2} - b c + a b} = x^{0} = 1$

$(i i) \sqrt[l m]{\frac{x^{l}}{x^{m}}} \times \sqrt[m n]{\frac{x^{m}}{x^{n}}} \times \sqrt[n l]{\frac{x^{n}}{x^{l}}} = (x^{l - m})^{\frac{1}{l m}} \times (x^{m - n})^{\frac{1}{m n}} \times (x^{n - l})^{\frac{1}{n l}} = x^{\frac{l - m}{l m}} . x^{\frac{m - n}{m n}} . x^{\frac{n - l}{n l}} = x^{\frac{l - m}{l m} + \frac{m - n}{m n} + \frac{n - l}{n l}} = x^{\frac{l n - m n + l m - l n + m n - l m}{l n m}} = x^{\frac{0}{l m n}} = x^{0} = 1$

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Similar questions

Q. Simplify:

x^{a + b} x^{b + c} x^{c + a}

Q. The coefficient of

x^{m}

{(1 + x)}^{m} + {(1 + m)}^{m + 1} + . . . + {(1 + x)}^{n}, m \leq n

State true or false.

a = x^{m + n} . y^{1}; b = x^{n + 1} . y^{m}

and

c = x^{1 + m} . y^{n}

, then

a^{m - n} . b^{n - 1} . c^{1 - m} = 1

Q. If

a = x^{m + n} \cdot y^{1}; b = x^{n + 1} \cdot y^{m} a n d c = x^{1 + m} \cdot y^{n}

,
then prove that

a^{m - n} \cdot b^{n - 1} \cdot c^{1 - m} = 1

Q. Simplify for

A

and

B

and find their sum, if

A = \frac{1}{1 + x^{b - a} + x^{c - a}} + \frac{1}{1 + x^{c - b} + x^{a - b}} + \frac{1}{1 + x^{a - c} + x^{b - c}}

and

B = {(\frac{x^{m}}{x^{n}})}^{1 / (m - n)} \times {(\frac{x^{n}}{x^{l}})}^{1 / (n - l)} \times {(\frac{x^{l}}{x^{m}})}^{1 / (l - m)}

then A+B

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Simplify: (i)(xa+bxc)a−b(xb+cxa)b−c(xc+axb)c−a(ii)lm√xlxm×mn√xmxn×nl√xnxl

Simplify: $(i) {(\frac{x^{a + b}}{x^{c}})}^{a - b} {(\frac{x^{b + c}}{x^{a}})}^{b - c} {(\frac{x^{c + a}}{x^{b}})}^{c - a} (i i) \sqrt[l m]{\frac{x^{l}}{x^{m}}} \times \sqrt[m n]{\frac{x^{m}}{x^{n}}} \times \sqrt[n l]{\frac{x^{n}}{x^{l}}}$