Question

simplify (x^a/x^b)^a^2+ab+b2 ×(x^b/x^c) ^b^2+bc+c^2×(x^c/x^a) ^c^2+ca+a^2​

Answer 1

Answer:

Step-by-step explanation:

Answer 2

Identities Used :-

$\boxed{\pink{\bf\: {a}^{x} \div {a}^{y} = {a}^{x - y}}}$

$\boxed{ \pink{ \bf \: {( {x}^{m}) }^{n} = {x}^{mn}}}$

$\boxed{\pink{\bf\:(x - y)( {x}^{2} + xy + {y}^{2}) = {x}^{3} - {y}^{3}}}$

$\boxed{ \pink{ \bf \: {x}^{0} = 1}}$

$\boxed{ \pink{ \bf \: {a}^{m} \times {a}^{n} = {a}^{m + n}}}$

Solution :-

$\sf \:{\bigg(\dfrac{ {x}^{a} }{ {x}^{b} } \bigg) }^{ {a}^{2} - ab + {b}^{2}}\times {\bigg(\dfrac{ {x}^{b} }{ {x}^{c}}\bigg) }^{ {b}^{2} + bc + {c}^{2}}\times {\bigg(\dfrac{ {x}^{c} }{ {x}^{a}}\bigg) }^{ {c}^{2} + ac + {a}^{2} }$

$\sf =\:{\bigg({x}^{a - b}\bigg) }^{ {a}^{2} - ab + {b}^{2}}\times {\bigg( {x}^{b -c } \bigg) }^{ {b}^{2} + bc + {c}^{2}}\times {\bigg( {x}^{c - a} \bigg) }^{ {c}^{2} + ac + {a}^{2} }$

$\sf =\:{\bigg({x}\bigg) }^{(a - b)({a}^{2} - ab+{b}^{2})}\times {\bigg({x}\bigg) }^{(b - c)({b}^{2}+bc+{c}^{2})}\times {\bigg( {x}\bigg) }^{(c - a)({c}^{2} + ac + {a}^{2})}$

$\sf \: = {x}^{ {a}^{3}-{b}^{3}} \times {x}^{ {b}^{3}-{c}^{3}} \times {x}^{ {c}^{3}-{a}^{3}}$

$\sf \: = \: {x}^{ {a}^{3}-{b}^{3}+{b}^{3} -{c}^{3}+{c}^{3}-{a}^{3}}$

$\sf \: = \: {x}^{0}$

$\sf \: = \: 1$

$\large\boxed{ \blue{ \bf \: Additional \: Information}}$

More Identities to know:

• (a + b)² = a² + 2ab + b²

• (a - b)² = a² - 2ab + b²

• a² - b² = (a + b)(a - b)

• (a + b)² = (a - b)² + 4ab

• (a - b)² = (a + b)² - 4ab

• (a + b)² + (a - b)² = 2(a² + b²)

• (a + b)³ = a³ + b³ + 3ab(a + b)

• (a - b)³ = a³ - b³ - 3ab(a - b)